The propagator and diffeomorphisms of an interacting field theory
Master thesis
submitted to the Institut f¨urPhysik Mathematisch-Naturwissenschaftliche Fakult¨at Humboldt-Universtit¨atzu Berlin
by
Paul-Hermann Balduf
in partial fulfillment of the requirements for the degree Master of Science.
Adlershof, July 5, 2018
Referees: Prof. Dr. Dirk Kreimer (supervisor) Dr. Christian Bogner We consider a scalar quantum field theory where the field φ is replaced by a diffeomor- 2 3 phism φ = ρ+a1ρ +a2ρ +.... The field ρ constitutes a modified quantum field theory defined implicitly by the diffeomorphism coefficients {aj}j and the Lagrange density of the underlying field φ. For a generic diffeomorphism, ρ is a non-renormalizable quan- tum field with infinitely many interaction vertices, even if φ itself is a free field. In the case that φ is an interacting field itself, ρ obtains additional vertices proportional to the coupling parameters in the Lagrangian density of φ. We examine the general Lagrangian density of a scalar field with local interactions,
∞ 1 µ 1 2 2 X λs s L = ∂µφ∂ φ − m φ − φ , 2 2 s! s=3 and show that the S-matrix elements of ρ coincide with the ones of φ. This implies the fields φ and ρ are indistinguishable in experiments. In this sense diffeomorphisms form equivalence classes of interacting scalar quantum field theories. On the other hand, n-point functions differ between φ and ρ if their momenta are offshell. Tuning the diffeomorphism coefficients {aj}j allows to change the behaviour of the offshell n-point functions of ρ. We use this freedom to eliminate all loop-corrections to the propagator of ρ for a fixed offshell four-momentum p. If φ has φs-type interaction,
1 µ 1 2 2 λ s L = ∂µφ∂ φ − m φ − φ , 2 2 s! and all tadpole graphs vanish, then the field formally given by λ ρ(x) = φ(x) − φs−1(x) (s − 1)!(p2 − m2) has a free propagator to all orders in perturbation theory, i.e.
ρ(p)ρ(−p) = −i p2 − m2.
Contents
1 Introduction5 1.1 Minkowski spacetime...... 5 1.2 Fuss-Catalan numbers...... 6 1.2.1 Generating function of A(2, 1)...... 7 1.2.2 Generating function of A(3, 1)...... 8 1.3 Bell polynomials...... 10 1.4 Axiomatic quantum field theory...... 11 1.4.1 Wightman axioms...... 11 1.4.2 Wightman distributions...... 12 1.4.3 Haag‘s theorem...... 13 1.5 Perturbative quantum field theory...... 14 1.5.1 Scattering theory in the interaction picture...... 14 1.5.2 Feynman graphs...... 17 1.5.3 Renormalization...... 18 1.6 Diffeomorphisms...... 21 1.7 Motivation for this work...... 21 1.8 Organization of the text...... 23
2 Free theory 25 2.1 Diffeomorphism of a free scalar theory...... 25 2.1.1 Lagrangian density...... 25 2.1.2 Feynman rules...... 27 2.1.3 Renormalizability...... 29 2.2 Tree sums...... 30 2.3 Multiple external offshell edges...... 33 2.4 Loop amplitudes...... 37
3 Phi3-theory 39 3.1 Diffeomorphism vertices of Phi3-theory...... 39 3.2 Cancellation of higher interaction vertices...... 41 3.2.1 Four external edges...... 41 3.2.2 General structure...... 42 3.2.3 Formal definition of Sn...... 43 3.2.4 Vanishing of Sn...... 46 3.3 Tree sums with interaction vertices...... 47 3.3.1 General form...... 47 3.3.2 Example: Tree sum with four external edges...... 48 3.3.3 Example: Tree sum with five external edges...... 51 3.4 Explicit cancellation of the 2-point function for one loop...... 57
3 3.5 Explicit calculation for two loops...... 60 3.5.1 Topology A...... 61 3.5.2 Topology B...... 62 3.5.3 Topology C...... 64 3.5.4 Topology D...... 68 3.5.5 Total result...... 72 3.6 All-orders cancellation of corrections...... 72 3.6.1 Decomposition of non-vanishing graphs...... 73 3.6.2 Factorization of tree sums...... 74 3.6.3 Parameters of the adiabatic diffeomorphism...... 79
4 Higher order interaction 83 4.1 Feynman rules...... 83 4.2 Tree sums and higher interaction vertices...... 85 4.3 Cancellation of corrections to the 2-point function...... 87 4.4 Multiple interaction monomials...... 94 4.4.1 Feynman rules...... 94 4.4.2 Tree sums...... 95
5 Properties of the adiabatic diffeomorphism 97 5.1 Rho as a free field...... 97 5.1.1 Rho in momentum space...... 97 5.1.2 Classical fields...... 98 5.1.3 Cancellation of higher correlation functions...... 99 5.2 An identity for PhiS-theory...... 99 5.3 Conclusion and outlook...... 101
Bibliography 103
A Lemmas 107 A.1 Sums of partitions of momenta...... 107 A.2 Compatibility of symmetry factors...... 109 A.3 Combinatoric Lemmas...... 111
B Umgangssprachliche Erl¨auterungder Arbeit 123
C Danksagung 127
D Selbstst¨andigkeitserkl¨arung 129
4 1 Introduction
1.1 Minkowski spacetime
Four-dimensional Minkowski spacetime M with speed of light c = 1 is used throughout. Four-vectors are underlined and three-vectors are bold such that the four-momentum is 0 1 2 3 p = p , p , p , p = (E, px, py, pz) = (E, p). The metric is flat and has negative sign, 1 0 0 0 0 −1 0 0 η = . 0 0 −1 0 0 0 0 −1 Four-dimensional spacetime indices are greek and summation is implicit,
2 µ µ ν 2 2 p = pµp = ηµνp p = E − p . (1.1) Other indices are not summed over unless explicitly denoted. The spacetime derivative ∂ is ∂µ = ∂xµ , especially the second derivative µ 2 ∂µ∂ = ∂0∂0 − ∂1∂1 − ∂2∂2 − ∂3∂3 = ∂t − 4 = . Physical particles (i.e. “really existent” as opposed to virtual particles used in interme- diate steps in computations) obey the relativistic energy-momentum-relation
E2 = p2 + m2. (1.2)
By eq. (1.1) the square of the four-momentum of a physical particle is its mass squared. In four-dimensional momentum space, eq. (1.2) forms a surface called the mass shell. Hence if a particle fulfills eq. (1.2) it is called onshell, otherwise offshell. Definition 1. For a four-momentum p assigned to a particle with mass m, the offshell parameter is defined as
2 2 xp = p − m . This parameter is zero if the momentum p belongs to a physical particle with mass m. The notation here follows slightly nontrivial rules: If the index is a letter like xp, this letter indicates the name of the corresponding four-momentum. If the index is a number, it is the running number of some (usually canonically) numbered momenta like x p2 − m2. 1 = 1 (1.3)
5 If sums and differences appear in the index, these indicate the corresponding operations with momenta, not indices themselves. This is, 2 x p p − m2, 1+2 = 1 + 2 (1.4) but generally x1+2 =6 x3. Note that xp only depends on the magnitude of the momen- tum, for any four-momentum p it is xp = x−p. The equation xp = xq does not imply p = q.
1.2 Fuss-Catalan numbers
Definition 2. For fixed a ∈ N0 and b ∈ N the Fuss-Catalan numbers are defined as b ma + b (ma + b − 1)! Am(a, b) = = b . ma + b m (ma + b − m)!m! Some useful values are
A0(a, b) = 1
Am(1, 1) = 1
Am(1, 2) = m b A (0, b) = . m m
Example 1: Fuss-Catalan numbers used in subsequent examples
The (ordinary) Catalan numbers {Cm}m∈N = {1, 1, 2, 5, 14, 42, 132, 429,...} are
1 2m (2m)! Am(2, 1) = Cm = = . (1.5) m + 1 m m!(m + 1)!
In the context of φ4-theory we encounter the choice a = 3, b = 1,
1 3m {Am(3, 1)}m∈N = = {1, 1, 3, 12, 55, 273, 1428, 7752 ...}. 2m + 1 m m∈N
Countless interpretations of the Fuss-Catalan numbers are known, see for example their entries A000108 (Am(2, 1)) or A001764 (Am(3, 1)) in the OEIS [Slo18]. Noteworthy, the Catalan number Cm is the number of different planar trees with n + 1 leaves built of 3-valent vertices. Similarly, Am(s, 1) counts the number of such trees if they are made of s-valent vertices. The generating function of the Fuss-Catalan numbers is defined by ∞ X m Ca,b(t) = t Am(a, b). (1.6) m=0
6 Lemma 1.1. Iff for q ∈ R a function R(t) obeys
R(t) − 1 = tRq(t) then for b ∈ R ∞ X b mq + b Rb(t) = tm . mq + b m m=0
Proof. This is [MSV06, theorem 2.1] with the replacement R(t) = 1 + w.
By lemma 1.1 the generating function eq. (1.6) of the Fuss-Catalan-numbers is defined equivalently via its functional equation
a b b Ca,b(t) = tCa,b(t) + 1 . (1.7)
From eq. (1.6) and A0(a, b) = 1 the boundary condition is
Ca,b(0) = 1. (1.8)
For b = 1 and a = 2 or a = 3 the functional equation can be solved explicitly.
1.2.1 Generating function of A(2, 1) Set a = 2, b = 1, this gives the generating function of ordinary Catalan numbers defined implicitly via eq. (1.7)
2 C2,1(t) ≡ C(t) = tC (t) + 1. (1.9)
Solving the quadratic equation, there are (for any given t) two solutions √ 1 ± 1 − 4t C (t) = . (1.10) 1/2 2t As can be seen in fig. 1.1 or computed explicitly, only one of them fulfills the boundary condition eq. (1.8) C(0) = 1, consequently the generating function of Catalan numbers is √ 1 − 1 − 4t C(t) = . (1.11) 2t
As a generating function according to eq. (1.6), C(t) = C2,1(t) has to be real-valued. This is the case for 1 t ≤ . 4
7 Figure 1.1: The two solutions Cj(t) from eq. (1.10) of the functional equation (1.9). Solid real part, dashed imaginary part. To distinguish them, the imaginary parts have been slightly shifted off the x-axis. Only C2(t) (red) fulfills the boundary condition C(1) = 1 from eq. (1.8).
1.2.2 Generating function of A(3, 1)
For a = 3, b = 1 set C(t) ≡ C3,1(t), then eq. (1.7) becomes the cubic equation
1 1 C3(t) − C(t) + = 0. (1.12) t t |{z} |{z} p q
It can be solved with the Cardanic formulae [Car68]. Define
p3 q 2 27t − 4 ∆ = + = 3 2 108t3 and two auxiliary functions
s r r q √ 3 1 −4 + 27t u(t) = 3 − + ∆ = − + , 2 2t 108t3 s r r q √ 3 1 −4 + 27t v(t) = 3 − − ∆ = − − . 2 2t 108t3
p 1 The complex cube roots have to be chosen to guarantee u · v = − 3 = 3t as shown in fig. 1.2. √ √ −1+i 3 −1−i 3 Introducing the cube roots of unity 1 = 2 and 2 = 2 , the three solutions
8 Figure 1.2: Functions u(t) and v(t). of eq. (1.12) are
C1(t) = u + v
C2(t) = 1u + 2v (1.13)
C3(t) = 2u + 1v.
Figure 1.3: The three solutions Cj(t) from eq. (1.13). The solution fulfilling eq. (1.8), C(1) = 1, can be constructed joining C1(t) and C2(t).
To fulfill the boundary condition eq. (1.8), the correct generating function is C1(t)
9 for t < 0 and and C(0) = 1 and C2(t) for t > 0. Explicitly, this is C (t) t < 0 1 C3,1(t) = 1 t = 0 (1.14) C2(t) t > 0 r q r q 3 1 27t−4 3 1 27t−4 − 2t + 108t3 + − 2t − 108t3 t < 0 = √ r q √ r q −1+i 3 3 1 27t−4 1+i 3 3 1 27t−4 2 − 2t + 108t3 − 2 − 2t − 108t3 t ≥ 0. The distinction of the two cases just means taking different cube roots. The generating function C3,1(t) has to be real-valued, this is the case only for 4 t ≤ . 27 1.3 Bell polynomials
We follow the definitions of [KY17]. See also [FG05, chapter 7] for details and proofs of the properties listed below.
Definition 3. For k ∈ N0, n ∈ N0 and k ≤ n, the Bell polynomials are
X n! x1 j1 x2 j2 x3 j3 xn jn Bn,k(x1, x2, x3,...) = ··· , j1!j2! ··· jn! 1! 2! 3! n! S where the sum runs over all {ji}i∈{1,...,k} such that
S = {ji ≥ 0 ∀i, j1 + j2 + j3 + ... + jk = k j1 + 2j2 + 3j3 + ... + (n − k)jn−k = n}. Special values are
B0,0 = 1
B0,k = 0, k > 0
Bn,0 = 0, n > 0 (1.15)
Bn,k = 0, k > n
Bn,1 = xn, n > 0 n Bn,n = x1 , n > 0. We will frequently need the generating function of Bell polynomials ∞ n n ∞ j X X k t X t Bn,k(x1, x2,...)u = expu xj . (1.16) n! j! n=0 k=0 j=1 Expanding the exponential function, exchanging summation order on the l.h.s. and extracting the coefficients of uk this becomes k ∞ ∞ X tn 1 X tj Bn,k(x1, x2,...) = xj . (1.17) n! k! j! n=k j=1
10 Bell polynomials have a striking combinatoric meaning: They count partitions of {1, . . . , n} into k nonempty disjoint subsets, i.e. X Bn,k(x1, x2, x3,...) = x|P1| ··· x|Pk|, P where
P = {∅= 6 Pi ⊆ {1, . . . , n} ∀i, Pi ∩ Pj = ∅ ∀i =6 j, P1 ∪ ... ∪ Pk = {1, . . . , n}}. (1.18) A consequence of eq. (1.16) is that Bell Polynomials represent the coefficients of composed diffeomorphisms. Fa`adie Bruno’s Formula [Wei18a] for the n-th derivative n ∂t of a composed function is n n X k 2 3 ∂t f(g(t)) = ∂t f (g(t)) · Bn,k ∂tg(t), ∂t g(t), ∂t g(t),... . k=0 P∞ tn P∞ tn In terms of power series f(t) = n=1 fn n! , g(t) = n=0 gn n! and ∞ X tn f(g(t)) =: h(t) = hn n! n=0 this becomes n X hn = fk · Bn.k(g1, . . . , gn+1−k). (1.19) k=1
1.4 Axiomatic quantum field theory
We consider scalar quantum fields exclusively, these are fields with spin zero. The following part follows Lutz Klaczynski’s PhD thesis [Kla16].
1.4.1 Wightman axioms The Wightman axioms are a mathematically rigorous way of defining quantum fields. They read 1. The states of the physical system are described by vectors in a separable Hilbert space h equipped with a strongly unitary representation (a, Λ) → U(a, Λ) of the ↑ connected Poincar´egroup P+ (i.e. the group of orthochronous proper Lorentz transformations and shifts). Moreover there is a unique vacuum Ψ0 ∈ h which is invariant under these Poincar´etransformations, U(a, Λ)Ψ0 = Ψ0. 2. The generator of the translation subgroup ∂ i U(a, 1) = pµ ∂aµ a=0
has its spectrum inside the closed forward light cone: σ(p) ⊂ V + and the generator of time translations (the Hamiltonian) has nonnegative eigenvalues, H = p0 ≥ 0.
11 3. For every Schwartz function f ∈ S(M) there are operators φ1(f), . . . , φn(f) (called † † quantum fields) and their adjoints φ1(f), . . . , φn(f) on h such that the polynomial algebra
D † E A(M) = φj(f), φj(f): j ∈ {1, . . . , n} C has a stable common dense domain D ⊂ h. This means
A(M)D ⊂ D ↑ U(a, Λ)D ⊂ D ∀(a, Λ) ∈ P+.
Further, the vacuum is cyclic for A(M), this means
Ψ0 ∈ D
D0 := A(M)Ψ0 ⊆ D is dense in h.
Finally, the maps
0 f → Ψ φj(f)Ψ
are tempered distributions on S(M) for all Ψ, Ψ0 ∈ D and j ∈ {1, . . . , n}.
4. The quantum fields transform under the unitary representation of the Poincar´e group according to
† U(a, Λ)φj(f)U (a, Λ) = φj({a, Λ}f)
on the domain D where
({a, Λ}f)(x) = f Λ−1(x − a)
is the Poincar´e-transformedtest function.
5. For f, g ∈ S(M) with mutually spacelike-separated support, this is
f(x)g(y) =6 0 ⇒ (x − y)2 < 0,
the commutator of quantum fields vanishes:
[φj(f), φl(g)] = φj(f)φl(g) − φl(g)φj(f) = 0 ∀j, l ∈ {1, . . . , n}.
1.4.2 Wightman distributions Wightman distributions are the n-fold vacuum expectation values of quantum fields
Wn(f1, . . . , fn) = hΨ0 φ(f1) ··· φ(fn)Ψ0i.
They fulfill a set of axioms equivalent to the Wightman axioms for the quantum fields themselves in the sense that given a set of Wightman distributions, there exists a
12 corresponding quantum field theory fulfilling the Wightman axioms (Wightman’s re- construction theorem, [Kla16, Theorem 10.1]). Especially, if φ is a free field without interaction, the 2-point distribution (where Ω0 instead of Ψ0 is used to stress the fact this state is the vacuum of a free theory) is Z 4 d p ∗ 0 2 2 hΩ0 φ(f)φ(h)Ω0i = f˜ (p)2πθ(p )δ(p − m )h˜(p), (1.20) (2π)4 where f˜ is the Fourier transform of f. If one takes the test functions f and h to have “sufficiently” small support around some points x, y such that they can be approximated by delta distributions
f(z) → δ(z − x), h(z) → δ(z − y) (1.21) then their Fourier transformations become
f˜∗(p) → e−ipx h˜(p) → eipy and one can use the notation
φ(x) := φ(f) f(y)≈δ(y−x)
φ(p) := φ(f˜) . f˜(p)≈exp(ipy)
Under this condition eq. (1.20) reduces to the function [Zwi17, eq. (7)], [Kla16, eq. (9.5)]
2 ∆+ x − y, m := Ω0 φ(x)φ(y)Ω0 (1.22) Z d4p = e−ip(x−y)θ(p0)δ(p2 − m2) (2π)3 1 Z d3p e−ip(x−y) = 3 p . 2 (2π) p2 + m2
This is a well-defined function for x =6 y but has a pole at x = y. This divergence indicates that the replacement eq. (1.21) is not allowed. The physical assumption that a quantum field be a well-defined operator for any sharp spacetime point x ∈ M is impossible to satisfy by non-singular quantum fields [Kla16, thm. 9.1].
1.4.3 Haag‘s theorem Two results severely restrict the behaviour of Wightman functions. Note that the Wightman functions eqs. (1.20) and (1.22) belong to free quantum fields, it is generally hard to compute a Wightman function of an interacting quantum field. A free 2-point function is the correlation function of a free quantum field according to eq. (1.20). Similarly, a free field has certain free n-point functions and by Wightman’s reconstruction theorem if all correlation functions coincide with the ones of a free field, the field itself is free of interaction.
13 The first theorem states that having a free 2-point function for a quantum field is sufficient that all higher n-point functions also coincide with the ones of a free field. Conversely, the this implies that no interacting quantum field can “by chance” have a free 2-point function and be otherwise interacting.
Theorem 1.1 Jost-Schroer theorem. If m ≥ 0 and φ is a scalar quantum field in Minkowski spacetime with the two-point 2 Wightman function of a free scalar field ∆+(x − y, m ) according to eq. (1.22), then φ is a free field with mass m.
Proof. [Kla16, thm. 11.6]. The case m > 0 is the original theorem by Jost and Schroer [Jos61], the case m = 0 was shown by Pohlmeyer in [Poh69].
The Jost-Schroer theorem underlines the importance of understanding the 2-point func- tion of any quantum field under consideration. The second theorem asserts that a free quantum field cannot be turned into an interacting one by an unitary transformation.
Theorem 1.2 Haag’s theorem. Let φ and φ0 be two Hermitian scalar quantum fields of equall mass m ≥ 0, the latter of which is a free field. Assume the sharp-time limits φ(t, f) and φ0(t, f) exist and form an irreducible set in their respective Hilbert spaces h and h0 for this specific time t = 0. If there is an isomorphism V : h0 → h (i.e. an unitary map) such that at t = 0
−1 φ(0, f) = V φ0(0, f)V then φ is a free field with mass m.
Proof. [Kla16, thm. 11.7]. Knowing the Jost-Schroer theorem, the proof reduces to showing that φ and φ0 have the same 2-point Wightman function.
1.5 Perturbative quantum field theory
1.5.1 Scattering theory in the interaction picture The perturbative treatment, althought its compatibility with axiomatic quantum field theory in some points appears unsatisfactory, is highly successfull at predicting exer- perimental observations and can be found in any QFT textbook. We follow [Das10; BD67] and also the overview given in [Kla16]
Definition 4. The time ordering operator Tˆ is for time-dependent operators φ(x), φ y defined as
Tˆφ(x)φ(y) = θ(x0 − y0)φ(x)φ(y) + θ(y0 − x0)φ(y)φ(x).
The operator with the latest time is arranged to the left position in the product. This readily generalizes to any product of operators.
14 The Feynman propagator is the time-ordered 2-point correlation function of the free scalar field and evaluates to
2 D E ∆F (x − y, m ) := Ω0 Tˆ[φ(x)φ(y)]Ω0 (1.23)
( 2 0 0 ∆+ x − y, m x > y = 2 0 0 ∆+ y − x, m x < y Z d4p e−ip(x−y) = lim . →0 (2π)4 p2 − m2 + i It is also the Green function of the Klein-Gordon operator acting on x, µ 2 2 ∂µ∂ + m ∆F (x − y, m ) = −δ(x − y). In the following, unless noted otherwise, the term n-point function will always denote time-ordered n-point vacuum expectation values like eq. (1.23) and not Wightman distributions like eq. (1.22). In the Heisenberg picture, the field operators φ(x) = φ(t, x) carry the full time evolution of the physical system. Conversely, in the Schr¨odingerpicture, the fields (like any operators) are constant in time φ = φ(x) and it is the states which change in time. The (time-independent) Hamiltonian H is the generator of time translations, hence
φ(t, x) = eiHtφ(x)e−iHt relates Heisenberg- and Schr¨odingerpictures if they agree at a time t = 0. The Hamil- tonian can be split into an interaction part and the Hamiltonian of the free field theory, H = Hint + H0. Then one define the interaction picture by
iH0t −iH0t φ0(t, x) = e φ(x)e . (1.24) Here φ(x) is the time-independent Schr¨odingerpicture field operator. The subscript φ0(t, x) is motivated by the fact that this field evolves in time by a free Hamiltonian, Z Z 3 3 1 2 1 1 2 2 H0 = d x H = d x π (x) + ∇φ(x)∇φ(x) + m φ (x) . 2 2 2 Combining both above definitions yields a relationship between Heisenberg picture and interaction picture,
iHt −iH0t iH0t −iHt φ(t, x) = e e φ0(t, x)e e † = V (t)φ0(t, x)V (t) (1.25) with a time evolution operator
V (t) = eiH0te−iHt. (1.26)
Hint = H − H0 is the time-independent interaction part of the Hamiltonian in the Schr¨odingerpicture. It is a polynomial in the fields φ(x), therefore in the interaction picture it becomes time-dependent as well,
† Hint,i(t) = V (t)HintV (t). (1.27)
15 For an evolution from time t1 to t2 in the interaction picture define
† iH0t2 −iHt2 iHt1 −iH0t1 U(t2, t1) = V (t2)V (t1) = e e e e Z t2 = Tˆ exp −i dτ Hint,i(τ) t1 The operator U(∞, −∞) =: S gives the time evolution from infinite past to infinite future and is called the S-matrix. This matrix encodes the probability for an initial state |ii to evolve from the infinite past to infinite future and reach a final state |fi by probability for evolution i → f = hf Sii. (1.28) Eventually, the goal of quantum field theory is to reproduce and predict experimental observations. In high energy physics, apart from masses of bound states, these are pri- marily cross-sections in scattering experiments. The Lehmann-Symanzik-Zimmermann reduction formula [LSZ55] relates a scattering process with n (both incoming and out- going counted) particles to n-point correlations functions of the underlying quantum field theory. Hence the main objective is to compute these correlation functions. The standard procedure to obtain perturbative approximations to n-point functions consists of three steps: First, if Ψ0 is the vacuum of H and Ω0 is the vacuum of the free Hamiltonian H0, then the time-ordered n-point functions of the Heisenberg- and interaction picture are related via the Gell-Mann-Low formula [GL51]
hΩ0 T [Sφ0(x1) ··· φ0(xn)]Ω0i hΨ0 T [φ(x1) ··· φ(xn)]Ψ0i = . (1.29) hΩ0 SΩ0i Next, the S-matrix is expressed via Dyson’s series ∞ n Z Z X (−i) 4 4 S = 1 + ··· d x ··· d x T [Hint(x ) ··· Hint(x )] (1.30) n! 1 n 1 n n=1 which reduces the n-point function of an interacting theory to an infinite sum of cor- relation functions of the free theory. Finally, Wick’s theorem [Wic50] is used to reduce a n-point function of the free theory to all possible products of 2-point functions, i.e. Feynman propagators eq. (1.23), of the free theory. Graphically, this can be depicted as a number of points connected with edges (which represent the propagators) such that the sum over all possible products equals the sum over all graphs fulfilling certain restrictions. These graphs are called Feynman graphs and the Wick theorem reads D E ˆ n1 nk X Y 2 Ω0 T [: φ0 (x1): ··· : φ0 (xk) :]Ω0 = c(Γ) i∆F (xstart(e) − xend(e), m ). Γ∈G e∈EΓ (1.31) Here, : ... : indicates normal ordering and c(Γ) involves several factors to be discussed in section 1.5.2. A detailled analysis shows that large classes of Feynman graphs can be ignored right away. Either they cancel against the denominator of eq. (1.29) or they can be recovered trivially as products of smaller graphs. Eventually it is sufficient to calculate Graphs which are 1PI according to the following definition:
16 Definition 5 (1PI). A connected graph is 1-particle-irreducible or 2-connected if any edge can be removed without decomposing the graph into two connected components wich are mutually disconnected.
1.5.2 Feynman graphs Definition of a Feynman graph A Feynman graph Γ formally consists of
1. a finite set of half-edges Γhe
he 2. a partition of Γ into a set of vertices VΓ
3. a set of disjoint unordered pairs of half-edges EΓ making up the internal edges.
The external edges of Γ are the half-edges not belonging to an internal edge e ∈ EΓ,
ext he Γ = Γ \ EΓ.
A Feynman graph automorphism is a bijection
f :Γhe → Γhe, respecting vertices v, inner edges e and external edges h,
v ∈ VΓ ⇒ f(v) ∈ VΓ
e ∈ EΓ ⇒ f(e) ∈ EΓ h ∈ Γext ⇒ f(h) = h.
The automorphisms form a group n o he he Aut(Γ) = f f is an automorphism Γ → Γ . Definition 6. The symmetry factor of a Feynman graph Γ is the number of automor- phisms it permits,
sym(Γ) = |Aut(Γ)|
Feynman rules So far, n-point functions have been correlation functions of the quantum field at dif- ferent points in spacetime. In scattering experiments on the other hand one observes the momenta of the particles rather than their position. To make the connection, the n-point functions are Fourier-transformed and computed in momentum space. Hence they are functions of n different momenta instead of n different spacetime points. Since Dyson’s series eq. (1.30) introduces “intermediate” spacetime points (i.e. points which are none of the xj the original n-point function is computed for), after Fourier trans- formation there are undetermined momenta in the Wick-expansion eq. (1.31). These momenta correspond to closed loops in the Feynman graph. By the general principles
17 of quantum mechanics, one needs to integrate over any unobserved quantity. Therefore a Feynman graph with loops contains one (4-dimensional) integral for each independent loop it has. The Feynman rules F are a mapping between Feynman graphs of the perturbation series eq. (1.31) and Feynman amplitudes as functions of the external momenta. If the graph Γ is a tree, F(Γ) is the product of the individual Feynman amplitudes of its constituents, i.e. vertex- and propagator amplitudes. If Γ contains loops, F(Γ) is an integral over all undetermined internal momenta and has the structure 1 Z N(Γ) Z F(Γ) = d4·|Γ|k = d4·|Γ|k I(Γ). (1.32) sym(Γ) Q x e∈EΓ e Here, I(Γ) was introduced as a shorthand for the Feynman integrand of Γ without the integration itself. I(Γ) is the product of the individual Feynman amplitudes of the constituents of Γ and depends on both external and integration momenta (and masses and coupling parameters). The number of independent loops (i.e. the first Betti number) of Γ is |Γ| and the numerator N(Γ) depends on the underlying theory. s For scalar φ -theory it involves a factor (−iλ) for each vertex in VΓ and a factor i for each internal edge in EΓ. This prescription follows from the Lagrangian of the theory. Especially, if there is a monomial λ − · φn n! λ n in the Lagrangian, there will be a corresponding monomial + n! · φ in the interaction Hamiltonian. Via the Dyson series eq. (1.30) such a monomial gives rise to a n-valent vertex with Feynman rule −iλ. Therefore, the factor c(Γ) in eq. (1.31) contains both 1 the symmetry factor sym(Γ) and the numerator N(Γ). Whenever an expression like Γ = f(p,...) appears, we implicitly mean F(Γ) = f(p,...), the Feynman rules F are the only mapping between graphs and functions appearing in this work. The integrand I(Γ) however will be denoted explicitly.
Euler characteristic The Euler characteristic [Wei18b] relates the number of edges, loops and vertices of a plane graph. If Γ is a plane connected graph and |EΓ| the number of its (internal) edges and |VΓ| the number of its vertices and then
|VΓ| − |EΓ| + |Γ| = 1 (1.33)
Especially, a tree does not contain any loops, hence a plane connected tree T has one more vertex than it has edges,
|VT | − |ET | = 1. (1.34)
1.5.3 Renormalization Feynman integrals as obtained from the Feynman rules eq. (1.32) are generally di- vergent. Technically the divergences arise from the ill-definedness of the replacement
18 eq. (1.21) to obtain pointlike quantum fields but can also be understood physically: In an interacting theory, there is self-interaction of the fields at any time. Doing a computation in the standard perturbative way as outlined in section 1.5.1 ignores this phenomenon by assuming the fields propagate as free fields between pointlike inter- action events. Therefore the divergences in unrenormalized quantum field theory are an indicator that the quantities computed there are meaningless because they ground on a misconception of interaction. Renormalization can be understood to correct this, see [Kla16, chapter 16.3]. A theory is called renormalizable if the superficial degree of divergence of a Feynman graph only depends on the number (and, if different types of fields are present, on the types) of external edges. For a renormalizable theory, the renormalization procedure eliminates all infinities in a consistent methodical manner. Motivated by classical (i.e. non-quantum) field theories, it seems plausible to write a Lagrangian density for an interacting φs theory as
1 µ 1 2 2 λ s L = ∂µφ∂ φ − m φ − φ . (1.35) 2 2 s! | {z } | {z } L0 Lint But to fully include the self-interactions of a quantized field, it turns out one has to use renormalized quantities s p 1 Z(λr) 2 Z(λr) φr = √ φ λr = λ mr = p m. (1.36) Z Zλ(λr) Zm(λr)
The Z-factors are themselves functions of λr. The free field (which does not need renormalization) is restored in the limit λ → 0 by the conditions Z(0) = 1,Zλ(0) = 1,Zm(0) = 1. Inserting eq. (1.36) turns eq. (1.35) into
1 µ 1 2 2 Zλλr s L = Z∂µφr∂ φr − Zmm φ − φ (1.37) 2 2 r r s! r 1 µ 1 2 2 λr s 1 µ 1 2 2 (Zλ − 1)λr s = ∂µφr∂ φr − m φ − φ + (Z − 1)∂µφr∂ φr − (Zm − 1)m φ − φ . 2 2 r r s! r 2 2 r r s! r | {z } | {z } Lr,0 Lr,int
The part Lr,0 equals the unrenormalized free Lagrangian L0 from eq. (1.35) with the only difference that φr and mr take the role of φ and m. On the contrary, Lr,int contains three additional free parameters Z(λr),Zm(λr),Zλ(λr) which formally give rise to further types of interactions, i.e. additional vertices in Feynman graphs. The renormalized interaction Hamiltonian density in the interaction picture (i.e. as a function of the interaction picture fields φr,0(t, x) as in eq. (1.27)) reads
1 µ 1 2 2 Zλλr s Hr,int(t, x) = − (Z − 1)∂µφr∂ φr + (Zm − 1)m φ + φ . 2 2 r r s! r Consequently, Dyson’s series eq. (1.30) also includes Z-factors. Since the Z-factors are functions of λr, ∞ ∞ ∞ X n X n X n Z = 1 + znλr Zλ = 1 + zλ,nλr Zm = 1 + zm,nλr , n=1 n=1 n=1
19 eventually Dyson’s series again has one single expansion paramter, namely λr. This procedure turns out to be feasible in many cases, especially in the standard model of particle physics. For a renormalizable theory it is possible to choose the coefficients zn, zλ,n and zm,n n in every order λr such that the Feynman integrals become finite. There are different possible renormalization schemes (eventually yielding the same physical result) [Col03]. We will use kinematic renormalization. Let Z F(Γ)(p) = d4k I(Γ)(p, k) be a logarithmic divergent Feynman integral without internal subdivergences, where p is an external momentum. Kinematic renormalization amounts to choosing some fixed p value 0 and computing the renormalized Feynman integral
Z Z F p, p 4k I p, k − I p , k 4k I p, p , k . R(Γ)( 0) = d (Γ)( ) (Γ)( 0 ) = d R(Γ)( 0 ) (1.38)
A graph with worse than logarithmic divergence behaviour requires the subtraction of derivatives of I(Γ), that is, the physical value of this quantity is fixed only up to some polynomial in the external momenta. In renormalizable theories, the degree of this polynomial matches the degree (in derivatives) of monomials in the Lagrangian density, allowing the ambiguity to be eliminated by finite redefinition of the renormalization factors Zi. If not more than two derivatives appear in L (as is the case for the Lagrangian eq. (1.37) we will be using), graphs may be at most quadratically divergent for the theory to be renormalizable. If the integral involves divergent sub-integrals, a single subtraction is not sufficient to render it finite. Instead, these subintegrals need to be renormalized first and then their renormalized version is used in the outer integral. But even then a renormalized integrand IR(Γ) can be written down explicitly, it consists of sums and products of differences of the individual subintegrands. The combinatorics of this procedure is fully understood and encoded in a Hopf algebra [Kre98; KW99]. Kinematic renormalization has the salient property that any Feynman integral which does not depend on external momenta is renormalized to zero. This can be understood from the simplest example eq. (1.38): In case the integrand I(Γ) does not depend on any external momentum, the very same integrand is subtracted and the renormalized integrand is zero. For a more complicated Feynman integral, it is sufficient if there is a single subintegral which does not involve external momenta for the whole integral to vanish. Such a subintegral arises as soon as there is a closed path of edges which is connected with the rest of the graph in just a single vertex. This shape is dubbed tadpole after the similar looking animal. It is possible and consistent with the Hopf algebra of renormalization to leave out any tadpole graph from the beginning. This way using kinematic renormalization signifi- cantly reduces the number of 1PI Feynman graphs to be considered for a given external leg structure and loop number. For later use, the following 1-loop integrals evaluate to zero in kinematic renormal-
20 ization because they belong to tadpole graphs :
4 Z d k ren. 1 −→ 0 (2π)2 4 Z d k 1 ren. 2 −→ 0 (1.39) (2π) xk 4 Z d k 1 ren. 2 −→ 0 for any fixed p. (2π) xk+p
For two loops, if u, v and w are arbitrary linear combinations of integration variables and external momenta,
ZZ 4 4 d kd l xu ren. 4 −→ 0. (1.40) (2π) xvxw
Essentially, the vanishing is due to the fact that if there are only two offshell variables x in the denominator, a linear transformation can always render them independent of external momenta. This is representing the graphical fact that every 2-loop graph which has only two internal edges is a tadpole due to the Euler characteristic section 1.5.2.
1.6 Diffeomorphisms
In general, a diffeomorphism is a bijection f between manifolds such that f and f −1 are differentiable. In our context of field theory, a diffeomorphism is a function f (not dependent on x) which relates one field ρ(x) to another field
φ(x) = f(ρ(x)) at the same spacetime point x. For this locality, diffeomorphisms are also called point transformations. In classical mechanics, point transformations are a subset of canonical transforma- tions of the generalized positions and momenta. In quantum theory, it seems point transformations are given by unitary operators but this operator has “a form of infinite series and, in general, will be never convergent and so never unitary as it is”[Nak55, p.384]. Therefore it is not at all clear that diffeomorphisms of quantum fields are a unitary transformation of these fields.
1.7 Motivation for this work
Perturbative scattering theory in the interaction picture in its most basic form as out- lined in section 1.5.1 relies on the assumption that for infinite past or future the states of an interacting quantum field theory resemble those of a free theory. In terms of scat- tering states |ii, |fi from eq. (1.28) this translates to the assumption that “sufficiently” far from the interaction point, particles propagate as free particles. This concept is usually introduced qualitatively in the form of physical intuition. One possible way is by using an interaction Hamiltonian which is switched on and off in
21 time, and goes under the name adiabatic hypothesis [Das10, p. 229]. But in the light of Haag’s theorem (1.2) it is questionable how an interacting and a non-interacting particle can possibly propagate in the same way. If their propagators, i.e. 2-point functions, coincide, both fields are necessarily free. Also the compatibility of Z-factors with the adiabatic hypothesis rises questions [BD67, sec. 16.4]. A second, probably even more severe problem is a core ingredient in the construction of the interaction picture, namely the assumption that for some time (in our case eq. (1.25) t = 0 has been chosen) the free fields of the interaction picture φ0 coincide with the interacting Heisenberg fields φ. Since Haag’s theorem dictates there is no unitary transformation between free and interacing fields, the operator eq. (1.26) cannot be unitary [Kla16]. Althought similar, these two problems are not the same: It is possible to solve the first problem by carefully establishing a notion of asymptotic states which for no time are unitarily related to free fields (Haag-Ruelle theory, an overview is given in [Fra06]). But this does not cure the second problem because the interaction picture relies on the relation eq. (1.25) no matter what the asymptotic states are. Existence of the interaction picture however is crucial for perturbative quantum field theory. Even if asymptotic states are used which do not violate Haag’s theorem, it is unclear how to obtain actual predictions for them if the machinery of Feynman graphs is not available. On the other hand, perturbative quantum field theory is highly successfull at predict- ing and reproducing experimental data. As one single example, consider the correction 1 ae = 2 (g − 2) of the magnetic moment g of the electron [HFG08]. The best currently −12 available experimental value [TP18] is ae,measured = (1159652180.91 ± 0.26) · 10 . An approximation of ae available from perturbative quantum field theory from 2012 −12 [Aoy+12] is ae,qft = (1159652181.13 ± 1.27) · 10 , it coincides with the experimental value within the given accuracy. The 4-loop integrals from quantum electrodynamics involved in ae,qft have been computed numerically to great precision recently [Lap17]. A possible remedy lies in the use of quantum field diffeomorphisms: As was demon- strated in [KY17], there is (at least formally) a diffeomorphism ρ(φ) of an interacting quantum field φ such that the propagator of ρ is free for a given fixed offshell momentum. At the same time, the S-matrix of ρ equals the one of φ. In this work, we will prove the latter statement and explicitly construct a suitable diffeomorphism for the former. In a profound analysis which is not pursued in this work, it might then be possible to refor- mulate the asymptotic conditions for scattering states in terms of this diffeomorphism and eliminate (contradictory) assumptions about the interaction Hamiltonian. For its implications regarding the adiabatic hypothesis, we call this special diffeomorphism “adiabatic”. It is so far completely unclear if the adiabatic diffeomorphism can also be utilized to approach the second problem, namely the construction of an “interaction picture” without unitary equivalence between free and interacting fields. A second motivation to study quantum field diffeomorphisms lies in the fact that these diffeomorphisms are generally perturbatively non-renormizable but still their S- matrix coincides with the one of a renormalizable theory. Since the diffeomorphism can be tuned arbitrarily, it might be possible to understand how to extract finite S- matrix elements even from a non-renormalizable theory in special cases. Especially, the degree of divergence of Feynman graphs for field diffeomorphisms is proportional to the loop number, this phenomenon also appears in the (not yet fully understood) quantum
22 theory of gravity, compare section 2.1.3. The third motivation for this work arises from path integrals [Fey48]. The path integral formalism is an alternative formulation of quantum field theory which we will not use here. A priori, the path integral is invariant under any diffeomorphisms of the field [ACO01], at least if corrections for non-commuting terms are taken into account [Omo77]. Therefore it is worth checking if and to what extent this invariance also exists in the canonical perturbative formulation of quantum field theory outlined in section 1.5. Finally, has been shown on the level of the Lagrangian density that the S-matrix is invariant under point transformations in renormalized perturbation theory in coordinate space [Flu75]. We consider Feynman graphs, i.e. the explicit terms arising in the perturbation series. Thereby we verify that the formal manipulations of the Lagrangian density are consistent with the perturbative expansion.
1.8 Organization of the text
This text is organized as follows: Chapter 2 reviews what is already known on diffeomorphisms of free quantum fields and fixes notation. The central result there is that diffeomorphisms of free fields do not alter the S-matrix, this implies they are not observable in scattering experiments. Chapter 3 represents the core part of this text, it discusses in great detail diffeomor- phisms of scalar φ3-theory. We establish that applying a diffeomorphism to the field does not change the S-matrix even if the underlying field has interaction. This allows to regard all quantum field theories which are related by diffeomorphisms as physically equivalent. Hence it is justified to pick a special field out of this class which has desir- able properties. We construct explicitly to all orders perturbation theory an “adiabatic diffeomorphism” which renders the time ordered offshell 2-point function free for a fixed momentum. Chapter 4 essentially repeats the steps from chapter 3 for a scalar theory with φs-type interaction. Again, diffeomorphisms do not alter the S-matrix and it turns out that the adiabatic diffeomorphism can be given explicitly even in this case. Appendix A is used to store a variety of lemmas which we need throughout the text. They are not given (and proved) in the main text because this would diminish readability. Appendix B is a summary of the whole thesis in colloquial german language. It is intended for the reader who has no background in physics and does not contain any additions relevant for the main text.
23
2 Free theory
Chapter 2 follows sections 3 and 4 of [KY17]. Note that the field ρ used in the following is what is called φ in [KY17]. An earlier investigation of free field diffeomorphisms is [KV13].
2.1 Diffeomorphism of a free scalar theory
2.1.1 Lagrangian density A free scalar quantum field theory is defined by its (unrenormalized) Lagrangian density
1 µ 1 2 2 L(x) = ∂µφ(x)∂ φ(x) − m φ (x). (2.1) 2 2 The first summand can be written in a different form using partial integration. This produces an additional surface term, but imposing either periodic boundary conditions or fields decaying at infinity eliminates this contribution and we arrive at
1 µ 1 2 2 L(x) = − φ(x)∂µ∂ φ(x) − m φ (x). (2.2) 2 2 The quantum field ρ is defined implicitly via the diffeomorphism ∞ 2 3 X j+1 φ(x) = ρ(x) + a1ρ (x) + a2ρ (x) + ... = ajρ (x), aj ∈ R. (2.3) j=0
We defined a0 = 1 and excluded a constant shift, this means the diffomorphism is tangent to the identity map. Inserting eq. (2.3) into eq. (2.2) yields the Lagrangian density for the field ρ:
∞ ∞ ! ∞ ∞ 1 X j+1 µ X k+1 1 2 X l+1 X m+1 L = − ajρ ∂µ∂ akρ − m alρ amρ 2 2 j=0 k=0 l=0 m=0 ∞ ∞ ∞ ∞ 1 X X j+1 µ k+1 1 2 X X l+1 m+1 = − aja ρ ∂µ∂ ρ − m a amρ ρ . (2.4) 2 k 2 l j=0 k=0 l=0 m=0 Under the spacetime integral to compute the action from the Lagrangian density, any total divergence vanishes and we can identify such terms with zero. The total divergence
µ n µ n µ n−1 0 = ∂µ(∂ ρ · ρ ) = ∂µ∂ ρ · ρ + n∂ ρ · ρ ∂µρ allows to set
µ n 1 n+1 µ ∂ ρ∂µρ · ρ = − ρ · ∂µ∂ ρ (2.5) n + 1
25 Using eq. (2.5), the summand in the first term of eq. (2.4) becomes
j+1 µ k+1 1 j+k+1 µ j+k+1 µ ρ ∂µ∂ ρ = (k + 1) − k ρ ∂µ∂ ρ + ρ ∂µ∂ ρ j + k + 1
(j + 1)(k + 1) j+k+1 µ = ρ ∂µ∂ ρ. j + k + 1 We rearrange summations in eq. (2.4) by introducing n = j + k + 1 instead of j and n = m + l + 2 instead of m: ∞ n−1 ∞ n−2 1 X X (n − k)(k + 1) n µ 1 2 X X n L = − a a ρ ∂µ∂ ρ − m a a ρ . (2.6) 2 n−k−1 k n 2 n−k−2 k n=1 k=0 n=2 k=0 The nested sums in the Lagrangian density of ρ can be written in a more systematic form by using Fa`adi Bruno’s formula eq. (1.19). The first derivative of φ is ∞ ∞ X n+1 X n 0 ∂µφ = an∂µρ = (n + 1)anρ · ∂µρ = φ · ∂µρ. n=0 n=0 Consequently, the kinetic term in eq. (2.1) amounts to
1 µ 1 µ 02 ∂µφ∂ φ = ∂µρ∂ ρ · φ . (2.7) 2 2 We write the outer derivative φ0 in a form where the constant contribution is extracted, ∞ n ∞ n 0 X ρ 0 X ρ 0 φ = (n + 1)!an = a0ρ + (n + 1)!an = 1 + ϕ . n! n! n=0 n=1 Now Fa`adi Bruno’s formula eq. (1.19) can be applied to compute (ϕ0)2. We set g(ρ) := 0 2 ϕ (ρ) and f(t) := t , i.e. gn = (n + 1)!an and fn = n!δn2, it follows ∞ n 0 2 X ρ f(g(ρ)) = ϕ (ρ) = 2 · Bn,2(2!a1, 3!a2, 4!a3,...) , n! n=1 2 2 φ0 = 1 + 2ϕ0 + ϕ0 ∞ X 2(n + 1)!an + 2Bn,2(2!a1, 3!a2 ...) = 1 + ρn. n! n=1 The first summand in the parenthesis can also be written in terms of Bell Polynomials
(n + 1)!an = Bn,1(2!a1, 3!a2,...) = Bn+1,1(1!a0, 2!a1, 3!a2,...), ∞ 2 X Bn+1,1(1!a0, 2!a1, 3!a2,...) + Bn,2(2!a1, 3!a2,...) φ0 = 1 + 2 ρn. (2.8) n! n=1 If eq. (2.5) is applied to eq. (2.8), the kinetic term eq. (2.8) takes the form ∞ 1 µ 1 µ X Bn,1(2!a1, 3!a2,...) + Bn,2(2!a1, 3!a2,...) µ n ∂µφ∂ φ = ∂µρ∂ ρ + ∂µρ∂ ρ · ρ 2 2 n! n=1 ∞ 1 µ X Bn−2,1(2!a1, 3!a2,...) + Bn−2,2(2!a1, 3!a2,...) n−1 µ = − ρ∂µ∂ ρ − ρ ∂µ∂ ρ. 2 (n − 1)! n=3
26 The mass term of eq. (2.1) can also be treated with Fa`adi Bruno’s formula eq. (1.19). We set g(ρ) = φ(ρ) and f(t) = t2, then
∞ n 2 X ρ f(g(ρ)) = φ(ρ) = 2 · Bn,2(1!a0, 2!a1, 3!a2,...) . n! n=2 The Lagrangian density of ρ is ∞ 1 µ X Bn−2,1(2!a1, 3!a2,...) + Bn−2,2(2!a1, 3!a2,...) n−1 µ L = − ρ∂µ∂ ρ − ρ ∂µ∂ ρ 2 (n − 1)! n=3 ∞ X Bn,2(1!a0, 2!a1, 3!a2,...) − ρn. (2.9) n! n=2
2.1.2 Feynman rules To obtain the vertex Feynman rules from eq. (2.9) we consider the two sums one after another. First, for any n ≥ 3, there is a term
Bn−2,1(2!a1, 3!a2,...) + Bn−2,2(2!a1, 3!a2,...) n−1 µ ρ ∂µ∂ ρ (2.10) (n − 1)!
In momentum space each derivative ∂µ becomes a momentum ipµ. Also, we sum over all permutations of the external legs, this gives a sum over the external momenta and for each summand a factor (n − 1)! for the possible permutations of the remaining (n − 1) legs. Finally, there is a factor i from the perturbation expansion eq. (1.30). The Feynman rule for eq. (2.10) is a n-valent Vertex with amplitude i B a , a ,... B a , a ,... p2 p2 ... p2 n−2,1(2! 1 3! 2 ) + n−2,2(2! 1 3! 2 ) 1 + 2 + + n dn−2 = i p2 + p2 + ... + p2 . 2 1 2 n
We have introduced a parameter dn consistent with [KY17],
dn = 2(Bn,1(2!a1, 3!a2,...) + Bn,2(2!a1, 3!a2,...)), n ≥ 1, d0 = 1. (2.11) If the explicit formula from definition 3 or the form eq. (2.6) is used, one obtains n X dn = n! an−kak(n − k + 1)(k + 1). (2.12) k=0 The second term in eq. (2.9) represents a n-valent vertex. Multiplying with the symmetry factor n! produces the Feynman amplitude
2 −iBn,2(1!a0, 2!a1, 3!a2,...) = im cn−2.
The factor cn is in accordance with [KY17], n 1 X cn = −Bn+2,2(1!a0, 2!a1, 3!a2,...) = − (n + 2)! a a . (2.13) 2 l n−l l=0
27 With the constants cn, dn, the Lagrangian density eq. (2.9) becomes
∞ ∞ 2 1 X dn−1 n µ X m cn−2 n L = − ρ ∂µ∂ ρ + ρ , (2.14) 2 n! n! n=1 n=2 that is, dn−1 and cn−2 are “coupling constants” for n ≥ 2-valent vertices. The n-valent vertex has the Feynman rule
dn−2 2 2 2 ivn = i p + ... + p + im cn−2. 2 1 n Instead of momenta, we want to use the offshell variables according to definition 1. Then
dn−2 2 2 ivn = i x1 + ... + xn + nm + im cn−2 2 dn−2 2 dn−2 = i (x1 + ... + xn) + im n + cn−2 2 2 2 = ifn · (x1 + ... + xn) + ignm . (2.15)
With eqs. (2.11) and (2.13) the coefficients fn, gn are given by
dn−2 fn = = Bn−2,1(2!a1, 3!a2,...) + Bn−2,2(2!a1, 3!a2,...) 2 n−2 (n − 2)! X = a a (n − k − 1)(k + 1), n ≥ 2, (2.16) 2 n−k−2 k k=0 and
dn−2 gn = n + cn−2 2 = n(Bn−2,1(2!a1, 3!a2,...) + Bn−2,2(2!a1, 3!a2,...)) − Bn,2(1!a0, 2!a1,...) n−2 n(n − 2)! X = a a (n − k − 2)k, n ≥ 2. (2.17) 2 n−k−2 k k=0
Example 2: Values of fn
1 f2 = 2 f3 = 2a1 2 f4 = 6a2 + 4a1 f5 = 24a3 + 36a1a2 2 f6 = 120a4 + 108a2 + 192a1a3.
28 Example 3: Values of gn
g2 = 0
g3 = 0 2 g4 = 4a1 g5 = 60a1a2 2 g6 = 288a2 + 432a1a3
iv3 = = 2ia1(x1 + x2 + x3)
2 2 2 iv4 = = 4im a1 + i 6a2 + 4a1 (x1 + x2 + x3 + x4)
2 iv5 = = 60im a1a2 + i(24a3 + 36a1a2)(x1 + x2 + x3 + x4 + x5)
. iv6 = .
Figure 2.1: Graphical representation of the n-valent vertices ivn produced by the diffeomorphism for n > . Their Feynman rules are given by eq. where x p2 − m2 are 2 (2.15) j = j the offshell variables of the corresponding legs. Note our conventions disagree with [KV13], there, white vertices were used for the massive part of vn alone whereas the part ∝ xj was denoted by black vertices.
Note that for n = 2 the Feynman rules eq. (2.15) reproduce the standard 2-vertex of a free theory, i.e. the Klein-Gordon-operator in momentum space
1 2 2 iv2 = i(xp + x−p) = ixp = i p − m . (2.18) 2 This is because the diffeomorphism eq. (2.3) is tangent to identity. But for any n > 2 there is a vertex with Feynman amplitude vn from eq. (2.15) which was not present in the original free theory φ. In Feynman graphs, we will draw those vertices by white (i.e. non-filled) dots as shown in fig. 2.1 to distinguish them from “real” vertices (caused by an interaction term in the Lagrangian density).
2.1.3 Renormalizability
2 Since vn ∼ p adds +2 to the degree of the integration variable while propagators give −2, by the Euler characteristic eq. (1.33), the superficial degree of divergence of any
29 graph Γ depends on the loop number only,
ω(Γ) = 4|Γ| + (−2)|EΓ| + (+2)|VΓ|
= 4|Γ| + (−2)(|VΓ| + |Γ| − 1) + (+2)|VΓ| = 2|Γ| − 2.
This implies that any non-tree graph is divergent, regardless of the external tree struc- ture, and for fixed number of external edges, the degree of divergence is non-constant. Thus, the field ρ represents a perturbatively non-renormalizable theory. Non-renorma- lizability is not even cured by the fact that Lρ has infinitely many monomials, thus technically allowing for infinitely many counterterms: Althought this suffices to absorb counterterms for all (infinitely many) divergent amplitudes up to a certain loop number, there are still infinitely many graphs with higher loop count which cannot be renormal- ized since they are worse than quadratically divergent. Renormalizing them requires the presence of terms with more than two derivatives in the Lagrangian density. Remarkably, the phenomenon of infinitely many divergent graph types with ω(Γ) = −2(|Γ| + 1), where precisely all 1-loop graphs are primitives in the renormalization Hopf-algebra, also appears in non-renormalizable quantum gravity [Kre08]. Therefore understanding field diffeomorphisms might be a step towards understanding a quantum theory of gravity.
2.2 Tree sums
A tree is a Feynman graph without loops, it represents a Feynman amplitude which does not involve any momentum integral but is merely a product of the Feynman amplitudes of the building blocks of the graph. For a given number of external edges a tree has the minimal possible number of vertices by the Euler characteristic section 1.5.2. Therefore in the perturbation series a tree always represents the lowest non-vanishing order in the coupling parameter.
Definition 7 (bn). If vn as in eq. (2.15) are the vertices arising from a diffeomorphism eq. (2.3) of a free quantum field theory then bn is the sum of all trees built from these vertices where the trees have n external onshell legs and precisely one external offshell leg. The propagator of the offshell leg is included in bn.
In Feynman graphs, the bn will be drawn as non-filled rectangles, see fig. 2.2. The onshell legs will be indicated with a perpendicular line at the end so if the edge carries momentum p, this means the corresponding offshell parameter is xp = 0.
30 b1 = = offshell edge, propagator included onshell edges, propagator not b2 = = included
P b3 = = + perm.
Figure 2.2: Sketch of contributions to the first tree sums bn. The external onshell edges are indicated with a bar at the end. For n ∈ {1, 2} there is only one vertex, vn+1, which can contribute. Starting from b3, there are also trees built from more than one vertex. Here “perm.” indicates a sum over all possible permutations of the external onshell edges.
Example 4: Explicit calculation of bn
For small n, the bn can be calculated explicitly. Let p be the offshell (upper in p , . . . , p fig. 2.2) momentum and let the onshell momenta be numbered 1 n. It is not quite possible to use eq. (2.18) for n = 1 because the lower edge being onshell also implies xp = 0 which makes the vertex vanish. Instead we define
b1 = 1. (2.19)
The next tree sum actually contains a vertex, namely i
b2 = 2ia1(x1 + x2 + x1+2) = −2a1. x1+2 x1=0,x2=0
For b3 we have to sum over v4 on the one hand and three different ways of assigning two v3 vertices on the other hand:
i 2 2 2 b3 = 4im a1 + i 6a2 + 4a1 x1+2+3 x1+2+3 i i + 2ia1(x1+2+3 + x1+2) 2ia1x1+2 x1+2+3 x1+2 i i + 2ia1(x1+2+3 + x1+3) 2ia1x1+3 x1+2+3 x1+3 i i + 2ia1(x1+2+3 + x2+3) 2ia1x2+3 x1+2+3 x2+3 2 2 m 2 2 2 x1+2 + x1+3 + x2+3 = −4a1 − 6a2 + 4a1 + 12a1 + 4a1 . x1+2+3 x1+2+3
31 0 2 Using lemma A.2, x1+2 + x2+3 + x1+3 = X4 = x1+2+3 + m , most terms cancel and
2 b3 = −6a2 + 12a1.
3 Similarly b4 = −24a3 + 120a1a2 − 120a1.
The bn can be defined recursively. They consist of all tree sums bs for s < n which are connected by a newly added upper vertex. Let there be k lower tree sums, then bn is produced by connecting these tree sums with a new vertex vk+1. The idea is shown in fig. 2.3. Then we have to sum over all possible partitions of the n onshell legs into k (non-empty) subsets. Finally, we have to sum over all k from one to n. So we are effectively summing over all non-empty, non-intersecting partitions of the n-element set of onshell momenta. These partitions are given by
P (k) = {P1 ∪ · · · ∪ Pk = {1, . . . , n}, |Pj| ≥ 1, ∀i =6 j : Pi ∩ Pj = ∅} (2.20)
Summing over these partitions, bn is defined recursively as n i X X bn = ivk+1 · b|P1| ··· b|Pk| (2.21) x1+2+...+n k=2 P (k)
Especially, the summand k = n consists of factors b1 = 1 only, this summand gives the biggest possible vertex vn+1 with no further vertices attached.
P P bn = = + ... + + P (3) P (2) ··· ··· | {z } n | {z } |{z} |{z} | {z }| {z } | {z } |P1| |P2| |P3| |P1| |P2| n
Figure 2.3: Sketch of contributions to a tree sum bn according to eq. (2.21). The tree sum consists of summands where an upper vertex of valence k + 1 is connected to k “lower” tree sums. Shown are the terms k = n (left) and k = 2 (right). The sums P (k) involve summation over the possibilities to divide n into k subsets (i.e. the valence of the lower tree sums) as well as the possible permutations of the onshell edges (i.e. which edge is connected to which of the lower tree sums).
Theorem 2.1. If bn is defined recursively as in eq. (2.21), then
n X (n + k)! bn+1 = B (−1!a1, −2!a2,..., −n!an). n! n,k k=1
Proof. This is proved in [KY17], Theorem 3.5. The proof starts by recognizing that P (k) from eq. (2.20) is essentially the same sum as in the definition of Bell polynomials
32 eq. (1.18), it is then possible to rewrite eq. (2.21) as a recursive identity for Bell poly- nomials. Then, the terms are split into one part porportional to m2 and another one p p i 6 j proportional to i j, = which is possible due to symmetry in the onshell momenta. After some manipulations and using lemma A.15, the individual terms boil down to lemma A.11.
By theorem 2.1, the objects bn are purely combinatorial. They do not depend on the involved momenta and masses. The tree sum with n + 1 external edges where all propagators are amputed arises from bn by multiplying the inverse upper propagator −ix1+2+...+n,
2 −i p p ... p − m2 b . 1 + 2 + + n n
If the upper edge is onshell as well, the parenthesis vanishes and so does the whole amplitude. This means that at treelevel the diffeomorphism does not contribute to the S-matrix. Compare [KV13, sec. 4] where this result is established without using theorem 2.1 by a discussion similar to the one in section 2.3 which will be useful several times in our study.
2.3 Multiple external offshell edges
In the tree sums from theorem 2.1, precisely one external edge is offshell and all others are onshell. In this section this result is generalized to more than one external edges offshell.
j Definition 8. An is the sum of all trees with n external edges, j of which are offshell.
We start with an example and then discuss the general phenomenon in quite some de- tail. This is because the same argument applies later in the discussion of an interacting theory and is therefore crucial for the results of this work. Essentially this argument is also layed out in [KY17, chapter 4.2] but in a more formal, abstract fashion. Following their proof would introduce a lot of notation overhead which will never be used later in this work, so here we pursue a more qualitative discussion instead. Consider the Feynman rule of the diffeomorphism vertex eq. (2.15):
2 ivn = ifn · (x1 + ... + xn) + ignm .
The fact that theorem 2.1 does not contain any terms proportional to m2 implies that in the sum over all trees and permutations, the contributions proportional to m2 are always cancelled. So it is safe to leave out all terms ∝ m2 (to be consistent, also the terms ∝ m2 generated through the mechanism lemma A.1 have to be left out) and assume vn consists of n summands where each is proportional to one of the adjacent offshell variables xj. Graphically, the vertex vn cancels each of its adjacent edges once and does not produce any constant terms.
33 An internal edge has two adjacent diffeomorphism vertices. So it can be cancelled either twice, or once, or not at all. We will consider these cases one at a time. The first case is what eliminates the m2 contributions of vertices through a mechanism like lemma A.1. Of course, there are also contributions of double-cancelled edges which 2 are proportional not to m but xj for some j. These contributions only change the proportionality constant for some already existing term, but not the fact if edges are cancelled or not. So, in terms of cancellation, a double-cancelled edge is an object which reduces to single-cancellation of its outer edges (and the double-cancelled edge itself is gone, so the double-cancelled edge together with its two adjacent vertices forms one new vertex which cancels one of its outer edges). Similarly, the edges cancelled once are already included in theorem 2.1, they do not do anything new here. In terms of Graphs, they cause their two adjacent vertices to fuse. A tree with v vertices has v−1 internal edges by the Euler characteristic section 1.5.2. Since each of the diffeomorphism vertices in tree sums cancels precisely one edge, hav- ing one external edge offshell (and cancelled) in bn implies that all internal edges are cancelled. Consequently, if more than one external edge is cancelled, an internal edge will remain uncancelled. So it is uncancelled internal edges which make the difference j between bn and some An+1 with j > 1 external offshell edges.
Example 5: Four external edges
4 We explicitly compute the Feynman amplitude A4 of a tree sum with four external edges, all of which may be offshell. There are two types of contributions, 1. one 4-valent vertex v4 and 2. two 3-valent vertices v3, connected with an internal edge. The latter involves three different ways of distributing the four external edges onto the two vertices. p , p , p , p Let the external momenta be 1 2 3 4. The 4-valent by eq. (2.15) vertex has amplitude
4 2 2 2 A4,1 = v4 = 4im a1 + i 4a1 + 6a2 (x1 + x2 + x3 + x4).
For the trees involving two vertices, first fix x1 and x2 to be the variables at one of the vertices and add the contributions with indices exchanged later.
4 i A4,2a = 2ia1(x1 + x2 + x1+2) 2ia1(x1+2 + x3 + x4) x1+2 2 x1 + x2 = −4ia1 + 1 (x1+2 + x3 + x4) x1+2 2 (x1 + x2)(x3 + x4) = −4ia1 x1 + x2 + x1+2 + + x3 + x4 x1+2
2 2 2 (x1 + x2)(x3 + x4) = −4ia1(x1 + x2 + x3 + x4) − 4ia1x1+2 − 4ia1 . x1+2
34 Summing over all three ways and using lemma A.1 eliminates the xi+j and yields 4 2 2 2 2 (x1 + x2)(x3 + x4) A4,2 = −16ia1(x1 + x2 + x3 + x4) − 4im a1 − 4ia1 + 2 more . x1+2
The total amplitude of a tree sum with four external edges is
4 4 4 A4 = A4,1 + A4,2 (2.22) 2 2 (x1 + x2)(x3 + x4) = i −12a1 + 6a2 (x1 + x2 + x3 + x4) − 4ia1 + 2 more . x1+2
Setting all xj except one to zero and including an external propagator in eq. (2.22) reproduces bn from theorem 2.1, i i 4 1 2 · A4 = · A4 = 12a1 − 6a2 = b3. x1 x2=x3=x4=0 x1
There is an important connection between an edge with momentum p being offshell and being cancelled:
Lemma 2.1. Let e be an edge adjacent to Γ where An is a sum over all possible trees with n external edges . If An contains vertices capable of cancelling adjacent edges, then either there are terms cancelling e or e is onshell.
Proof. Assume the edge e is actually onshell. Then by definition 1, xe = 0. The parts of the integrand An which can cancel e are precisely the summands proportional to xe since cancellation amounts to multiplication with the inverse propagator, which is −ixe. But since xe = 0, all these summands vanish. This means there exists no term in the integrand which can cancel the edge, hence it is uncancelled. On the contrary, since there is at least one vertex in An generating a factor xj for its adjacent edge j, by symmetry also a term xe must appear in the sum over all trees. So if this term does not vanish then it will cancel e.
Lemma 2.1 seems trivial, it is nevertheless the core ingredient for constructing tree sums with uncancelled internal edges. Assume for simplicity there are two external offshell edges, so just one internal edge e is uncancelled. Then from the point of view of its adjacent subtrees this edge e is an onshell edge. Nevertheless it is not actually onshell, this is, there is still a propagator i with x =6 0. So the whole tree disintegrates into xe e two components which are connected via this edge. Summing over all trees especially includes summing over all ways to reorganize these components without changing the uncancelled edge. But these are precisely the sums which yield bs by definition 7. This 2 is, a tree sum An with two of its n external edges offshell consists of two tree sums bs1 , bs2 such that s1 + s2 = n and they are connected by an internal edge e. This edge has a propagator i where p is the sum of all momenta entering either of the two parts. xe e In summing over all permutations of external edges, this becomes a symmetric function 2 of the possible intermediate momenta as in eq. (2.22). Since An also is symmetric in
35 external edges, the remaining momentum-dependent function connecting bs1 and bs2 is symmetric in both the external momenta and the sums of external momenta which make up the uncancelled internal propagator. But since the latter appear in the denominator, it is not a symmetric polynomial in xj but rather a rational function. One might be tempted to reject this procedure because, apart from edges being can- celled or not, a Feynman amplitude generally is a function of its external momentan and therefore it makes a difference if these momenta are onshell or not. This is true indeed but does not apply here: The “building blocks” of the tree sum are by con- struction precisely the parts where no internal edge is uncancelled. They are always tree sums where the sum is carried out without any restriction on the allowed internal cancellations, i.e. some bn according to definition 7. But by theorem 2.1, no bn depends on external momenta. One might also object that for bn, all but one external momenta are onshell and in this sense the amplitude does depend on the specific value of the momenta. This is also true, but if more then one external edge were offshell, still the terms forming bn would be present, there would just be additional summands which are proportional to some xj. In the end the whole argument reduces to recognizing that with diffeomorphism Feynman rules eq. (2.15) there cannot be any other dependence on external momenta than in the form of terms proportional to some xj. Consequently, if all terms proportional to (some power of) xj are zero then the whole tree sum is necessarily independent of xj and hence of the corresponding external momentum. Consider for illustration the computation example 5 where from the very beginning 4 x4 = 0 is imposed. The result would be just the same as calculating A4 for arbitrary x4 and setting x4 = 0 in the end result, i.e. all terms which are not proportional to x4 are unchanged no matter if x4 is onshell or not.
Example 6: Four external edges The result eq. (2.22) from the above example 5 can be interpreted in terms of tree sums bs connected by symmetric functions of the external momenta. (x1 + x2)(x3 + x4) A4 = −i(x1 + x2 + x3 + x4) · b3 − i + 2 more · b2 · b2. x1+2 (2.23)
36 Example 7: Five external edges A computation similar to example 5 yields for the tree sum with five external edges
3 A5 = i −120a1a2 + 24a3 + 120a1 (x1 + x2 + x3 + x4 + x5) (2.24) 3 (x1 + x2)(x3 + x4 + x5) + i 24a1 − 12a1a2 + 9 more x1+2 3 (x1 + x2)x3(x4 + x5) + 8ia1 + 14 more . x1+2x4+5
Again, this expression consists of factors bs connected with internal propagators:
A5 = −i(x1 + x2 + x3 + x4 + x5) · b4 (x1 + x2)(x3 + x4 + x5) − i + 9 more · b3 · b2 x1+2 (x1 + x2)x3(x4 + x5) − i + 14 more · b2 · b2 · b2. x1+2x4+5
In the last line, 2 + 2 + 2 = 6 =6 5. This is because this line represents terms where three external edges are offshell (the symmetric part is of order three in the xj), hence there are two uncancelled internal lines. That the values s of bs add up to 6 then follows from the Euler characteristic eq. (1.34) regarding bs as meta-vertices with valence s + 1.
2.4 Loop amplitudes
There are two different approaches to see that even for loop amplitudes the diffeo- morphism does not contribute to the S-matrix. First by the Euler characteristic sec- tion 1.5.2. A connected graph Γ with |VΓ| vertices and |Γ| loops has |VΓ|+|Γ|−1 internal edges. Each diffeomorphism vertex effectively cancels one edge. So if no external edge is cancelled, i.e. if all external edges are onshell, then |VΓ| internal edges are cancelled. Each cancellation identifies two vertices, so the remaining graphs have the topology of a rose (i.e. a single vertex with several tadpoles) with |Γ| − 1 loops. Since this is a tadpole, it vanishes in kinematic renormalization. Even if one external edge is offshell, still all vertices are identified and the graph becomes a rose. Theorem 2.2. (l) If Am is a connected l-loop amplitude with m external edges where not more than one (l) is offshell then in kinematic renormalization Am = 0
Proof. This is [KY17, thm 4.7]. The above argument is given there, especially in lemma 4.6.
37 If two or more external edges are offshell and hence cancelled, the graph no longer shrinks to a rose. Clearly, to identify all |VΓ| vertices, at least |VΓ| − 1 internal edges have to be cancelled. But as soon as at least two external edges are cancelled, the total number of available cancellations, |VΓ|, is not sufficient for this. Hence at least two vertices remain distinct. This allows for the existence of graphs which do not vanish in kinematic renormalization. That is, if at least two of the external momenta are offshell, the diffeomorphism does alter the n-point functions. Especially, if precisely two external edges are offshell then all non-vanishing graphs will have the form of multiedges on two vertices. Note this only holds under the condition that eq. (2.15) are the only vertices present, i.e. only if the underlying theory is free. An alternative way to obtain vanishing of loop amplitudes is from Cutkosky rules [BK15; Zwi17]. The imaginary part of Feynman amplitudes is determined by inte- grals where some of the internal propagators are replaced by delta functions 2πδ(p2 − m2)θ(p0). This amounts to setting these edges onshell, i.e. the integrand then reduces to the product of two integrands where the corresponding edges are onshell, multiplied by these delta functions. The complete Feynman amplitude can then be recovered from its imaginary part using dispersion relations. Especially, it is zero if the imaginary part vanishes. By repeated application of this procedure, the uncut parts eventually reduce to trees. But if at most one external edge is offshell, at most one of these trees has a nonzero amplitude and hence any product between trees vanishes. This in turn means the imaginary part of the overall amplitude is zero. The actual procedure how many and which edges have to be cut to obtain the correct imaginary part is non-trivial but it is not necessary to examine it in detail here because it is clear that any product of two tree amplitudes will yield zero. The result that loop amplitudes do not contribute to the S-matrix is a direct consequence of the fact that tree amplitudes do not. Again, this argument breaks down for n-point functions with more than one offshell momentum since then there are at least two non-vanishing trees, hence there are products of trees which do not evaluate to zero. It might seem that Cutkosky rules circumvent the necessity for kinematic renormal- ization, i.e. vanishing of tadpoles. This is however not true since one needs the full amplitudes to be cut-reconstructible which might not be the case for other renormal- ization schemes [KV13, sec. 5.2].
38 3 φ3-theory
3.1 Diffeomorphism vertices of φ3-theory
We apply the diffeomorphism
∞ X j+1 φ(x) = ajρ (x), (3.1) j=0 which again is tangent to identity, to an interacting theory with Lagrangian density
1 µ 1 2 2 λ 3 L (x) = − φ(x)∂µ∂ φ(x) − m φ (x) − φ (x). (3.2) φ 2 2 3!
The field ρ contains the diffeomorphism vertices vn with Feynman rules eq. (2.15) as well as another type of vertex which arises from the interaction term in the Lagrangian density. This term is
∞ ∞ ∞ λ 3 λ X X X j+k+l+3 − φ = − aja a ρ 3! 3! k l j=0 k=0 l=0 ∞ n−3 n−3−j λ X X X n = − aja a ρ . 3! k n−3−j−k n=3 j=0 k=0
Hence, for any n ≥ 3 there is a vertex with Feynman rule
n−3 n−3−j λ X X −iwn = −i n! aja a 3! k n−3−j−k j=0 k=0
= −iλhn. (3.3)
The parameter hn introduced therein explicitly reads
n−3 n−3−j n! X X hn = aja a . 3! k n−3−j−k j=0 k=0
39 Example 8: Values of hn
h3 = 1
h4 = 12a1 2 h5 = 60a2 + 60a1 3 h6 = 360a3 + 720a1a2 + 120a1.
As with propagator of the free theory in eq. (2.18), the tangential diffeomorphism reproduces the usual 3-valent interaction vertex of φ3-theory
−iw3 = −iλ.
In terms of Feynman rules, the object wn can be understood as a generalized coupling parameter for an n-point interaction. Like eq. (2.14), the Lagrange density for ρ can be written
∞ ∞ 2 ∞ 1 X dn−1 n µ X m cn−2 n X wn n L = − ρ ∂µ∂ ρ + ρ − ρ . (3.4) 2 n! n! n! n=1 n=2 n=3
To distinguish the new diffeomorphism-interaction vertices −iwn from the pure dif- feomorphism vertices vn from eq. (2.15), the interaction-diffeomorphism vertices wn are drawn black as in fig. 3.1. The 3-valent vertex is a small dot. This is the only remaining vertex if the diffeomorphism is the identity, i.e. if aj = 0 for all j ≥ 1.
−iw3 = = −iλ
−iw4 = = −12iλa1
2 −iw5 = = −60iλ a2 + a1
. −iw6 = .
Figure 3.1: Graphical representation of the n-valent vertices wn produced by the diffeomorphism applied to a φ3 theory with Lagrange density eq. (3.2). The Feynman rules are eq. (3.3). Since these vertices arise from an actual interaction term, they are depicted black. The 3-valent vertex itself is smaller than the higher vertices to distinguish it.
40 3.2 Cancellation of higher interaction vertices
In [KY17, Theorem 5.1], the following claim is made for φ4-theory: The interacting theory is diffeomorphism invariant: the n-point interaction λ of order λ vanishes for n > 4 and is 4! for n = 4. No graph-theoretic proof apart from the first nontrivial example (which is the equivalent 4 of the example considered in section 3.2.1, just S5 for φ -theory) is given there. To prove the statement, it might be possible to reconstruct all higher orders from this first example using Hopf-algebraic techniques like in [KV13]. But in the following, we will use an approach similar to the treatment of the free field in chapter 2. First, we show that tree sums do not contribute to the S-matrix. Then, by the same argument as in section 2.4, also the loop amplitudes vanish for onshell external edges.
3.2.1 Four external edges As an introductory example of the mechanism and notation, the treelevel 4-point func- tion of the diffeomorphism of φ3-theory is computed explicitly here. To this end one needs to sum all trees with four external edges, built of both diffeomrphism- and interaction-vertices. Actually, there is one type of tree consisting only of the origi- nal 3-valent vertices w3 eq. (3.3). These vertices are the same as if no diffeomorphism was applied and so are the resulting trees. Therefore it is sensible to concentrate on the corrections which arise from the diffeomorphism, namely the 4-valent interaction vertex w4 and trees containing one diffeomorphim vertex v3 and one interaction vertex w3. The sum of these two contributions is dubbed S4. Finally, there are trees consisting of only diffeomorphism-3-vertices v3, but these tree sums without any interaction vertex have been discussed in section 2.3, especially eq. (2.23).
P S4 = + perm. | {z } S4,1 | {z } S4,2a
Figure 3.2: The two contributions to S4. Left the 4-valent interaction vertex −iw4 from fig. 3.1, right a tree sum consisting of one vertex −iw3 and one diffeomorphism 2-tree sum b2. The latter coincides with the diffeomorphism-3-vertex iv3 as shown in fig. 2.2. “perm.” is the sum over all six ways of assigning the four external edges to the two vertices.
The two contributions to S4 are depicted in fig. 3.2. The first one is just a single vertex with amplitude eq. (3.3),
S4,1 = −iw4 = −12iλa1, the second one has, for a fixed assignment of external edges, the amplitude
S4,2a = −iλ · 1 · 1 · (−2a1) = 2iλa1.
41 The term S4,2a is independent of momenta, it just needs to be multiplied by the number of possible permutations of external legs, which is the number of ways to choose two elements out of four, 4 = 6. 2 The total amplitude then is
S4 = S4,1 + 6S4,2a
= −12iλa1 + 6(2iλa1) = 0. (3.5) Note that this does not imply the 4-point function to be zero altogether. On the one hand there are still the trees consisting entirely of 3-valent interaction vertices −iλ like in ordinary φ3-theory, on the other hand for offshell 4-point functions there is a contribution from pure diffeomorphism, eq. (2.23), which is independent of the presence of any interaction term in the original Lagrangian density. Therefore S4 = 0 just says that the onshell tree sum of ρ has the same amplitude as the onshell tree sum of the original φ.
3.2.2 General structure Two observations determine the way a cancellation of higher order interaction-vertices −iwn from eq. (3.3) (with n > 3) can work:
1. Any interaction vertex −iwn is of order one in lambda. This implies that a cancellation of such a vertex for n > 3 can only possibly occur against tree sums which are also of this order. These tree sums must contain precisely one 3 interaction vertex, either an original φ vertex −iw3 = −iλ or a higher interaction vertex −iwm with 3 < m < n. All other vertices of the tree sum can only be of pure diffeomorphism type, i.e. be ivs from eq. (2.15) for some s.
2. The interaction vertex −iwn does not cancel adjacent edges. Therefore its am- plitude does not change if these edges are onshell. Hence one can assume that all edges are onshell. The corresponding tree sums generally do depend on the offshell variables of their external edges. Consequently, the cancellation has to work with the part of the tree sum amplitude which is independent of external momenta. This is precisely the part where all external offshell variables are set to zero which equals the edges being onshell i.e. the part where no external edges are cancelled (compare the similar argument in section 2.3). A pure diffeomorphism tree sum with k external onshell edges and one edge offshell is just bk from theorem 2.1. To fulfill both requirements, the tree sums cancelling −iwn have to consist of one interaction vertex which is connected to an arbitrary number of diffeomorphism tree sums bkj where the offshell edge of bkj connects to the interaction vertex and k1 + ... = n. Figure 3.3 shows the principle. Sn consists of the interaction vertex −iwn itself and then all possible ways to distribute the external edges into different bk. In terms of combinatorics, this resembles the construction of bn itself in fig. 2.3.
42 P Sn = + ... + ··· P (3) | {z } |{z} |{z} | {z } |P1| |P2| |P3| n
Figure 3.3: Structure of the contributions to Sn: Since the external edges are onshell, all terms consist of tree sums bk and one suitable interaction vertex wj. The sum P (k) runs over all possible ways of distributing the external edges to the given number k of tree
sums B|kj |. As opposed to fig. 2.3 there is no P (2) here because there is no 2-valent interaction vertex.
3.2.3 Formal definition of Sn
The formal definition of Sn, the sum of all contributions to the n-point onshell treelevel amplitude at order λ1, follows the construction sketched in fig. 3.3. First recall from eq. (2.19) that b1 = 1. Hence any external edge which directly connects to the inter- action vertex (without any bk in between) can formally be assigned a b1. Introducing these factors has the advantage that every edge of the interaction vertex is connected to a tree sum bk now, where possibly k = 1. Contributions to Sn include the vertex −iwn itself, connected to n “tree sums” b1, with the amplitude
−iwn b1b1 ··· b1 . | {z } n factors
The second term in Sn is a vertex −iwn−1 connected to one b2 and (n − 2) tree sums of type b1,
−iwn−1b2 b1b1 ··· b1 . | {z } n−2 factors
The third term is proportional to the vertex −iwn−2. In this case there can either be one b3 and (n − 3) b1, or there are two b2 and (n − 4) b1, giving rise to the amplitude
−iwn−2b2b2 b1b1 ··· b1 −iwn−2b3 b1b1 ··· b1 . | {z } | {z } n−4 factors n−3 factors
This scheme continues until the interaction vertex is just −iw3 and three tree sums bk1 , bk2 , bk3 are connected such that kj ≥ 1, k1 + k2 + k3 = n. The last summand in Sn has the amplitude
−iw3bk1 bk2 bk3 .
For each summand of Sn one additionally needs to sum over all possibilities to assign the n external edges to the bkj . Fix k ∈ {3, . . . , n} and consider the term where the
43 interaction vertex has valence k. This term is a product of −iwk and some factors bj but there is an additional sum over partitions of external edges:
k X Y Sn,k = −iwk bki . P (k) i=1
We have to request ki ≥ 1, therefore P (k) is the sum over partitions of a n-element set into k nonempty disjoint parts,
P (k) = {|Pi| = ki ≥ 1, k1 + ... + kk = n, P1 ∪ ... ∪ Pk = {1, . . . , n},Pi ∩ Pj = ∅}.
This partition P (k) is the very same set as in the definition of the Bell polynomials eq. (1.18) (for a fixed k) , hence
k X Y Sn,k = bki = Bn,k(b1, b2,...). P (k) i=1
Sn,k is the contribution to Sn where the interaction vertex is of valence k. To obtain Sn, one has to sum over all k ∈ {3, . . . , n} and
n n X X Sn = Sn,k = − iwkBn,k(b1, b2,...). (3.6) k=3 k=3
1 With the Feynman rule of wk from eq. (3.3) the sum of all contributions ∝ λ to the treelevel n-point amplitude is
n k−3 k−3−r λ X X X Sn = −i k!B (b1, b2,...) arasa . (3.7) 3! n,k k−3−r−s k=3 r=0 s=0
Since Sn in eq. (3.7) is just proportional to λ, the cancellation, if it works at all, cannot depend on a relationship between λ and the diffeomorphism coefficients {ai}i.
44 Example 9: S4
The four-point amplitude S4 was computed explicitly in section 3.2.1. Indeed, with
2 B4,3(x1, x2, x3, x4) = 6x2x1 4 B4,4(x1, x2, x3, x4) = x1. this result is reproduced from the general formula eq. (3.7):
4 k−3 k−3−r λ X X X S4 = −i k!B (b1, b2,...) arasa 3! 4,k k−3−r−s k=3 r=0 s=0 0 0−r λ X X = −i 3!B4,3(b1, b2,...) arasa0−r−s 3! r=0 s=0 1 1−r λ X X − i 4!B4,4(b1, b2,...) arasa1−r−s 3! r=0 s=0
= −iλ6b2 − 12iλa1
= −iλ6(−2a1) − 12iλa1 = 0.
The cancellation thus works just as in eq. (3.5). As claimed, it is valid without any special relationship between λ and the diffeomorphism coefficients {ai}i.
Example 10: S5
Also S5 vanishes:
5 X k! X S5 = −iλ B (b1, b2,...) asatau 5,k 3! k=3 s+t+u=k−3 3! X 4! X = −iλB5,3(b1, b2,...) asatau − iλB5,4(b1, b2,...) asatau 3! 3! s+t+u=0 s+t+u=1 5! X − iλB5,5(b1, b2,...) asatau 3! s+t+u=2 2 2 3 5 2 = −iλ 15b1b2 + 10b1b3 − iλ10b1b212a1 − iλb120 3a2 + 3a1 2 2 = −15iλb2 − 10iλb3 − 120iλb2a1 − 60iλa2 − 60iλa1 2 2 2 = −15iλ(−2a1) − 10iλ(−6a2 + 12a1) − 120iλ(−2a1)a1 − 60iλa2 − 60iλa1 = 0.
45 3.2.4 Vanishing of Sn Theorem 3.1. With eq. (3.7), it is Sn = 0 for any n > 3.
Proof. The formula eq. (3.7) depends on the tree sums {bk}k which in turn depend on the parameters {ak}k via theorem 2.1. Apply lemma A.15 in the form n−k X (n − 1 + j)! B (b1, b2,...) = B (−1!a1, −2!a2,...) 0 < k < n (3.8) n,k (k − 1)!(n − k)! n−k,j j=0 to transform it into an identity involving ak only. Since k < n is required, the last sum- mand in eq. (3.7) has to be considered independently. Further, the constant prefactors in Sn are irrelevant for the proof, so define 3! s = S n iλ n n−1 k−3 k−3−r X X X = − k!Bn,k(b1, b2,...) arasak−3−r−s k=3 r=0 s=0 n−3 n−3−r X X − n!Bn,n(b1, b2,...) arasan−3−r−s. r=0 s=0 n Putting in eq. (3.8) and using Bn,n(b1, b2,...) = b1 = 1 yields n−1 n−k k−3 k−3−r X X (n − 1 + j)! X X sn = − k! B (−1!a1, −2!a2,...) arasa (k − 1)!(n − k)! n−k,j k−3−r−s k=3 j=0 r=0 s=0 n−3 n−3−r X X − n! arasan−3−r−s. (3.9) r=0 s=0 An explicit calculation of the last summand k = n reveals 0 X (n − 1 + j)! (n − 1)! B0,j(−1!a1, −2!a2,...) = B0,0(−1!a1, −2!a2,...) = 1. (n − 1)!0! (n − 1)! j=0 Therefore this summand can be taken into the sum eq. (3.9), n n−k k−3 k−3−r X X (n − 1 + j)! X X sn = − k B (−1!a1, −2!a2,...) arasa . (3.10) (n − k)! n−k,j k−3−r−s k=3 j=0 r=0 s=0
Now substitute −j!aj = xj, the starting condition a0 = 1 then becomes x0 = −1 and
n n−k k−3 k−3−r 3 X X (n − 1 + j)! X X (−1) xrxsxk−3−r−s sn = − k! B (x1, x2,...) (k − 1)!(n − k)! n−k,j r!s!(k − 3 − r − s)! k=3 j=0 r=0 s=0 n k−3 k−3−l n−k X X X k xlxmxk−3−l−m X = (n − 1 + j)!B (x1, x2,...). (n − k)! l!m!(k − 3 − r − m)! n−k,j k=3 l=0 m=0 j=0 (3.11) For n =6 3, this vanishes by lemma A.10.
46 Theorem 3.1 is only about tree sums with one single interaction vertex, i.e. tree sums proportional to λ1. But tree sums with more than one interaction vertex automatically vanish, too. This is shown in the following section 3.3. Effectively, theorem 3.1 implies that even if the underlying theory has a cubic interaction, the diffeomorphism still does not contribute to the S-matrix at treelevel and, by section 2.4, also not at any loop order.
3.3 Tree sums with interaction vertices
3.3.1 General form
In theorem 2.1, the tree sums bn from definition 7, consisting exclusively of diffeomor- phism vertices, have been computed. Now for the diffeomorphism of φ3-theory there are additional vertices −iwn from eq. (3.3) which contribute to tree sums. Call these 0 tree sums bn: 0 0 Definition 9. bn bn is the sum of all trees, including both pure diffeomorphism and interaction-diffeomorphism vertices, with n external onshell edges and additionally one external offshell edge where the propagator is included. 0 There always is a tree sum in bn where all vertices are of diffeomorphism type ivk, 0 therefore bn = bn +.... Theorem 3.1 severely restricts the influence of higher interaction vertices to tree sums: Assume there is a vertex −iwn for n > 3 in the tree sum. There are two cases:
1. All (internal or external) edges connected to −iwn are non-cancelled. Then, in the sum over all possible trees, there will be precisely all trees at the position of −iwn to make up Sn. But this sum vanishes due to theorem 3.1. Hence all vertices −iwn where no adjacent edge is cancelled can be left out from the beginning. However, Sn also involves −iws for 3 ≤ s < n, where the edges adjacent to −iws are cancelled (compare fig. 3.3). Even these subtrees need to be left out.
2. At least one adjacent edge to −iwn is cancelled. This cancellation always origi- nates from a diffeomorphism vertex, not from −iwn itself, since it does not cancel anything. But then, −iwn and the adjacent vertex form a contribution to Sm for some m > n. Since Sm = 0, even these contributions can be left out. 3 Now all that remains are vertices −iw3 = −iλ, which are the original φ -vertices, where all three adjacent edges are non-cancelled. From the point of view of tree sums, an internal edge connecting to −iw3 is non- cancelled and hence acts as an onshell-edge, compare the discussion in section 2.3. Lemma 3.1. 0 bn consists of pure diffeomorphism tree sums bk for 2 < k < n which are connected (via symmetric sums of internal propagators) with their onshell edges to an arbitrary number 0 of interaction vertices −iw3. Just the smalles tree sum b2 cannot be decomposed this way, it is
0 λ b2 = −2a1 + . (3.12) xp
47 Consider the examples eqs. (3.15) and (3.16) below as an illustration of this lemma.
Proof. Follows from the above discussion. Especially, the second term −λ = i (−iw ) xp xp 3 0 in b2 is a tree with interaction vertex and one edge (with offshell variable xp =6 0) offshell.
0 An important consequence of this is that, unlike bn, the bn do depend on masses and momenta via the internal uncancelled propagators. But since neither −iw3 nor bk depend on masses and momenta, the sum over all permutations of external edges collapses and does not introduce any symmetric polynomials of xj, just symmetric terms in some 1 . xi+j+... The sum over all tree topologies will produce several terms, all of which are propor- tional to some bk (with different values of k). Especially Lemma 3.2. 3 0 In a diffeomorphism of φ -theory with tree sums bn with n external onshell edges and one edge offshell, including all interaction vertices wn:
0 0 bn vanish for all kinematic configurations ⇔ bk = 0 ∀k ≤ n, k > 2 and b2 = 0.
0 Proof. Note b2 = 0 involves the external variable xp, so it depends on external momenta. Then, “all kinematic configurations” means all possible configurations such that the offshell edge has the variable xp (which does not fix the fourvector p uniquely) and the onshell edges are onshell (which fixes only the magnitude, but not the direction of their momentum) and overall momentum is conserved. So there is freedom for different kinematic configurations even if xp and x1 = 0, . . . , xn = 0 is fixed. The direction ⇐ is a direct consequence of lemma 3.1. The direction ⇒ also follows from that lemma since all kinematic configurations are allowed. Otherwise, it would be possible to choose momenta (and thus internal momenta) in such a way that sums over internal propagators vanish.
0j Tree sums with more than one external leg offshell, A n, work precisely as in sec- tion 2.3 regardless of the presence of interaction vertices: The more external edges are offshell (i.e. cancelled), the more internal edges remain uncancelled. The following two comprehensive examples sections 3.3.2 and 3.3.3 illustrate this mechanism as well as lemma 3.1.
3.3.2 Example: Tree sum with four external edges 4 In example 5 the tree sum A4 with four (arbitrary offshell) external edges was computed. 04 4 Here A 4, a similar object including interaction vertices, will be considered. Since A4 is known, the trees consisting only of diffeomorphism vertices are left out. Ten different trees as shown in fig. 3.4 remain.
48 1 1 1 2 e 2 e 2 e Γ = Γ = Γ = 4 1 4 2 3 4 3 3 3
1 1 1 2 e 2 e 2 e Γ4 = 4 Γ5 = 4 Γ6 = 4 3 3 3
1 1 1 2 e 2 e 2 e Γ = Γ = Γ = 4 7 4 8 3 4 9 3 3
2 1
Γ = 10 3 4
04 Figure 3.4: Contributions to the tree sum A 4. The trees consisting of only diffeomorphism vertices are not shown.
49 The Feynman amplitude of the first type is i Γ1 = −iλ 2ia1(xe + x3 + x4) xe x3 + x4 = 2iλa1 + 2iλa1 xe and hence x3 + x4 x1 + x4 Γ1 + ... + Γ6 = 12iλa1 + 2iλa1 + 4 more + . x3+4 x1+4
Due to momentum conservation x1+2 = x3+4 there are only three different denominators which share the same numerator,
Γ1 + ... + Γ6 = 12iλa1 x1 + x2 + x3 + x4 x1 + x2 + x3 + x4 x1 + x2 + x3 + x4 + 2iλa1 + + . x1+2 x1+3 x1+4 The trees with two interaction vertices give i Γ7 = (−iλ) (−iλ) 1+2 2 1 1 1 Γ7 + Γ8 + Γ9 = −iλ + + . x1+2 x1+3 x1+4
Finally, Γ10 is a vertex −iw4 = −12iλa1 and thus
Γ1 + ... + Γ9 + Γ10 1 1 1 = iλ(λ − 2a1(x1 + x2 + x3 + x4)) + + . x1+2 x1+3 x1+4 The total tree sum including diffeomorphism eq. (2.23) and interaction is
04 4 A 4 = Γ1 + ... + Γ10 + A4 1 1 1 = −iλλ −2a1(x1 + x2 + x3 + x4) + + | {z } x1+2 x1+3 x1+4 b2 2 − i 12a1 − 6a2 (x1 + x2 + x3 + x4) | {z } b3 2 (x1 + x2)(x3 + x4) − i 4a1 + 2 more . (3.13) |{z} x1+2 b2·b2
All terms are proportional to some symmetric polynomials of the xj except one. Note the highest order of these polynomials is two (e.g. the term (x1 + x2)(x3 + x4)), this is because there are at most two vertices and each vertex can cancel at most one external edge. If one sets all xj = 0 (i.e. if all external legs are onshell), one gets 00 04 2 1 1 1 A 4 = A4 = −iλ + + . (3.14) onshell x1+2 x1+3 x1+4
50 This is the expected treelevel S-matrix element for φ3-theory, i.e. theorem 3.1 is fulfilled. Setting x = 0, x = 0, x = 0 in eq. (3.13) and including the propagator i gives the 2 3 4 x1 0 tree sum b3 where three external edges are onshell. It is i i 0 01 04 b3 = A 4 = A 4 x1 x1 x2=0,x3=0,x4=0 λ 1 1 1 = λ − 2a1 + + x1 x1+2 x1+3 x1+4 2 + 12a1 − 6a2 ,
0 Identifying b2 from eq. (3.12), this is 0 0 1 1 1 b3 = b3 + λb2 + + . (3.15) x1+2 x1+3 x1+4
Clearly, lemma 3.1 is fulfilled.
3.3.3 Example: Tree sum with five external edges
Γ1 = Γ2,m = Γ2,r =
Γ3,m = Γ3,r = Γ4 =
Γ5,1 = Γ5,2 = Γ6 =
05 Figure 3.5: General structure of trees contributing to A 5. As opposed to fig. 3.4, this figure shows only the general shape and type of vertices in a tree, but not the permuta- 04 tions of external edges. Note that unlike for A 4, the vertices here are generally not interchangable, e.g. Γ2,m is not just some permutation of legs of Γ2,r.
The tree sums with five external legs allow for three general structures as shown in fig. 3.5: There can be one, two or three vertices. In the latter case, if all three vertices are equivalent as in Γ1, there are 15 different possibilities to assign the outer edges. We first fix the central edge to be number 5 and one of the pairs edges number 1 and 2. In
51 a second step, the other 14 contributions will be added. With this convention,
3 1 Γ1 = iλ . x1+2x3+4
If one of the vertices is of pure diffeomorphism type,
2 1 Γ2,m = 2iλ a1(x5 + x1+2 + x3+4) . x1+2x3+4
In the case Γ2,r the diffeomorphism vertex can be one of two equivalent ones, the sum of both amplitudes is
2 1 Γ2,r = 2iλ a1(x1 + x2 + x1+2 + x3 + x4 + x3+4) . x1+2x3+4
The sum of all 3-vertex amplitudes with precisely one diffeomorphism vertex is 2 1 2 1 1 Γ2 = 2iλ a1(x1 + x2 + x3 + x4 + x5) + 4iλ a1 + . x1+2x3+4 x1+2 x3+4
The same procedure for the trees with two diffeomorphism vertices yields
2 1 Γ3,m = 4iλa1(x1 + x2 + x1+2)(x3 + x4 + x3+4) x1+2x3+4 2 (x1 + x2) (x3 + x4) 2 (x1 + x2) 2 (x3 + x4) 2 = 4iλa1 + 4iλa1 + 4iλa1 + 4iλa1. x1+2 x3+4 x1+2 x3+4
2 1 Γ3,r = 4iλa1(x5 + x1+2 + x3+4)((x1+2 + x1 + x2) + (x3+4 + x3 + x4)) x1+2x3+4 2 1 1 = 4iλa1(x1 + x2 + x3 + x4 + x5) + x1+2 x3+4 2 1 2 2 x1+2 2 x3+4 + 4iλa1x5(x1 + x2 + x3 + x4) + 8iλa1 + 4iλa1 + 4iλa1 . x1+2x3+4 x3+4 x1+2
The sum of all trees with three vertices of which at least one is diffeomorphism reads
2 2 1 Γ1 + Γ2 + Γ3 = 12iλa1 + iλ λ + 2λa1(x1 + x2 + x3 + x4 + x5) x1+2x3+4 2 1 2 x1+2 2 x3+4 + 4iλa1x5(x1 + x2 + x3 + x4) + 4iλa1 + 4iλa1 x1+2x3+4 x3+4 x1+2 1 1 + 4iλa1(λ + a1(x1 + x2 + x3 + x4 + x5)) + x1+2 x3+4
2 (x1 + x2) (x3 + x4) 2 x1 + x2 2 x3 + x4 + 4iλa1 + 4iλa1 + 4iλa1 . x1+2 x3+4 x1+2 x3+4
52 Adding the other 14 permutations of external edges, this becomes 2 2 1 1 = 180iλa1 + iλ (λ + 2a1(x1 + x2 + x3 + x4 + x5)) + + ... x1+2x3+4 x1+3x2+4 2 1 1 1 2 + 4iλa1x5(x1 + x2 + x3 + x4) + + + ... + 4iλa1x1(...) x1+2x3+4 x1+3x2+4 x1+4x2+3 1 1 1 1 1 1 + 4iλa1(λ + a1(x1 + x2 + x3 + x4 + x5)) + + + + + + ... x1+2 x3+4 x1+2 x2+5 x1+2 x3+5 2 (x1 + x2) (x3 + x4) (x1 + x3) (x2 + x4) + 4iλa1 + + ... x1+2 x3+4 x1+3 x2+4 2 x1 + x2 x1 + x2 x1 + x2 x1 + x3 + 4iλa1 + + + + ... x1+2 x1+2 x1+2 x1+3 2 x3 + x4 x3 + x4 x3 + x4 + 4iλa1 + + + ... x3+4 x3+4 x3+4 2 x1+2 x1+5 x2+5 x3+4 x3+5 + 4iλa1 + + + ... + + + ... x3+4 x3+4 x3+4 x1+2 x1+2 2 x3+4 x3+5 x4+5 + 4iλa1 + + + ... . x1+2 x1+2 x1+2 If one pair of external edges is fixed, there are precisely three possibilities to order the remaining three edges into one pair and a single edge. This is indicated in the third line: The summand 1 appears three times, each time with a different second summand. x1+2 Lines 7 and 8 as well as lines 5 and 6 can be added, each one then contains 30 summands. But since there are just 10 possibilities to choose a pair out of five elements, three summands have to coincide. 2 2 1 1 = 180iλa1 + iλ λ + 2λa1(x1 + x2 + x3 + x4 + x5) + + ... x1+2x3+4 x1+3x2+4 2 1 1 1 + 4iλa1x5(x1 + x2 + x3 + x4) + + x1+2x3+4 x1+3x2+4 x1+4x2+3 2 1 + 4iλa1x1(x2 + x3 + x4 + x5) + ... − ... x2+3x4+5 1 1 + 12iλa1(λ + a1(x1 + x2 + x3 + x4 + x5)) + ... + x1+2 x4+5 2 (x1 + x2) (x3 + x4) (x1 + x3) (x2 + x4) + 4iλa1 + + ... x1+2 x3+4 x1+3 x2+4 2 x1 + x2 x1 + x3 x1 + x4 + 12iλa1 + + + ... x1+2 x1+3 x1+4 2 x1+2 + x1+5 + x2+5 x3+4 + x4+5 + x3+5 + 4iλa1 + ... + + ... . x3+4 x1+2 By a computation similar to lemma A.1 one finds 2 x1+2 + x1+5 + x2+5 = x1 + x2 + x5 + x3+4 + m , x1+2 + x1+5 + x2+5 x1 + x2 + x5 1 = 1 + + m2 . x3+4 x3+4 x3+4
53 There are ten terms of this kind,
Γ1 + Γ2 + Γ3 2 2 1 1 = 220iλa1 + iλ (λ + 2a1(x1 + x2 + x3 + x4 + x5)) + + ... x1+2x3+4 x1+3x2+4 2 1 1 1 + 4iλa1x5(x1 + x2 + x3 + x4) + + x1+2x3+4 x1+3x2+4 x1+4x2+3 2 1 + 4iλa1x1(...) + ... + ... x2+3x4+5 2 1 1 + 4iλa1 3λ + 3a1(x1 + x2 + x3 + x4 + x5) + a1m + ... + x1+2 x4+5 2 (x1 + x2) (x3 + x4) (x1 + x3) (x2 + x4) + 4iλa1 + + ... x1+2 x3+4 x1+3 x2+4 2 x1 + x2 x1 + x3 x1 + x4 + 12iλa1 + + + ... x1+2 x1+3 x1+4 2 x1 + x2 + x5 x3 + x4 + x5 + 4iλa1 + ... + + ... x3+4 x1+2
Next consider the trees consisting of a 4-vertex and a 3-vertex. There are 10 permu- tations of external edges. First fix 1 and 2 to be the external edges connected to the 3-vertex. i Γ4 = 12iλa1 iλ x1+2
2 1 2 2 2 1 Γ5,1 = −2iλ(2a1 + 3a2)(x3 + x4 + x5) − 2iλ(2a1 + 3a2) − 4im λa1 x1+2 x1+2
2 1 2 Γ5,2 = −24iλa1 (x1 + x2) − 24iλa1. x1+2
One needs to sum over all permutations, i.e. the edges at the 3-vertex can be (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5). 2 1 1 Γ4 + Γ5 = −4iλa1 3λ + m a1 + ... + x1+2 x4+5 2 x3 + x4 + x5 x1 + x2 + x3 − 2iλ(2a1 + 3a2) + ... + x1+2 x4+5 2 x1 + x2 x4 + x5 2 − 24iλa1 + ... + − 280iλa1 − 60iλa2. x1+2 x4+5
Finally, the 5-vertex contributes
2 Γ6 = 60iλ a1 + a2 .
54 Total sum: 0 2 1 1 A5 = A5 + iλ (λ + 2a1(x1 + x2 + x3 + x4 + x5)) + + ... x1+2x3+4 x1+3x2+4 2 1 1 1 + 4iλa1x5(x1 + x2 + x3 + x4) + + x1+2x3+4 x1+3x2+4 x1+4x2+3 2 1 + 4iλa1x1(...) + ... + ... x2+3x4+5 2 (x1 + x2) (x3 + x4) (x1 + x3) (x2 + x4) + 4iλa1 + + ... x1+2 x3+4 x1+3 x2+4 2 x3 + x4 + x5 x1 + x2 + x3 + 6iλ(2a1 − a2) + ... + . x1+2 x4+5
This is symmetric in the xi, but there are different ways to write the terms, for example lines 2, 3 and 4 can be combined to terms like P x1x3 + x1x4 + x1x5 + x2x3 + x2x4 + x2x5 + x3x5 + x4x5 i Inserting eq. (2.24), the tree sum eventually becomes 05 3 A 5 = i −120a1a2 + 24a3 + 120a1 (x1 + x2 + x3 + x4 + x5) 3 (x1 + x2)(x3 + x4 + x5) + i 24a1 − 12a1a2 + 9 more x1+2 3 (x1 + x2)x3(x4 + x5) + 8ia1 + 14 more x1+2x4+5 2 1 1 + iλ (λ + 2a1(x1 + x2 + x3 + x4 + x5)) + + 13 more x1+2x3+4 x1+3x2+4 P P 2 i 55 This time, there are polynomials up to order three in xj since there are up to three vertices. As for eq. (3.13), setting all offshell variables to zero fulfills theorem 3.1. Setting all variables except x to zero and multiplying with i extracts the tree sum 1 x1 0 b4: i i 0 01 05 b4 = A 5 = A 5 x1 x1 x2=0,x3=0,x4=0,x5=0 3 = 120a1a2 − 24a3 − 120a1 2 λ 1 1 − λ + 2a1 + + 13 more x1 x1+2x3+4 x1+3x2+4 2 1 1 − λ(12a1 − 6a2) + 4 more + x2+3 x4+5 The prefactors are indeed bk as stated in lemma 3.1, 0 1 1 1 1 1 1 b4 = b4 − λb3 + + + + + x2+3 x2+4 x2+5 x3+4 x3+5 x4+5 2 0 1 1 + λ b2 + + 13 more (3.16) x1+2x3+4 x1+3x2+4 0 0 and setting b2 = 0, b3 = 0 and b4 = 0 implies b4 = 0 as proposed in lemma 3.2. 56 3.4 Explicit cancellation of the 2-point function for one loop We have established that a diffeomorphism does not change the S-matrix even in φ3- theory. We approach the second topic of this work: Constructing a specific “adiabatic” diffeomorphism to make the 2-point function free at a given fixed external momentum p. Since the diffeomorphism is tangent, it reproduces the free propagator at lowest order eq. (2.18). It is sufficient that all 1PI corrections to the 2-point functions vanish. Recall that in renormalizable theories, using a kinematic renormalization scheme makes all tadpole graphs vanish. Althought ρ is not renormalizable, we assume that all tadpole graphs can be left out consistently even in this case. In the present section, we determine suitable diffeomorphism coefficients ai by explic- itly writing down all Feynman integrals and requiring their sum to vanish. This section as well as the next one (section 3.5) serves mostly as a comprehensive example of the mechanisms at work. A discussion for the all-order procedure follows afterwards. k k Γ = Γ = 1 p −p 3 p −p k + p k + p k k Γ2,1 = Γ2,2 = p −p p −p k + p k + p Figure 3.6: 1-loop-graphs contributing to the 2-point function of a diffeomorphism of φ3-theory. The interaction vertex iλ is a small black dot, the diffeomorphism vertex a white dot as in fig. 2.1. All edges are bosons, the arrowhead indicates the direction the four-momentum is counted in. Since all tadpole graphs vanish in kinematic renormalization, the only 1-loop graphs with two external edges are multiedges with two internal edges between two 3-valent vertices. The diffeomorphism of φ3-theory has two different 3-valent vertices: the origi- 3 nal φ -interaction-vertex with Feynman amplitude iw3 = iλ from eq. (3.3) as well as the 3-valent diffeomorphism vertex iv3 = 2ia1(x1 + x2 + x3) from eq. (2.15). Consequently, there are four different ways of assigning these vertex types to the two vertices in the multiedge, giving rise to four different Feynman graphs which contribute to the 2-point function at 1-loop level. They are shown in fig. 3.6. We choose k to be the integration variable. 1 By definition 6, the 1-loop multiedge has a symmetry factor of 2 due to the inter- changeability of the internal edges. Therefore the Feynman integrands of the graphs in fig. 3.6 are 57 1 2 i i I(Γ1) = (−iλ) 2 xk xk+p 1 i i I(Γ2,1) = (−iλ)2ia1(xk + xk+p + xp) 2 xk xk+p 1 i i I(Γ2,2) = (−iλ)2ia1(xk + xk+p + xp) 2 xk xk+p 1 2 i i I(Γ3) = (2ia1) (xp + xk + xk+p)(xp + xk + xk+p) . 2 xk xk+p The graphs Γ2,1 and Γ2,2 give rise to the same integrands as can be expected from symmetry, we add those two contributions I(Γ2) = I(Γ2,1) + I(Γ2,2). (3.17) After expanding the parentheses the integrands read 1 2 1 1 I(Γ1) = λ 2 xk xk+p 1 1 1 1 I(Γ2) = −2λa1 − 2λa1 − 2λa1xp xk+p xk xk xk+p 2 2 1 1 1 1 xk xk+p I(Γ3) = 2a1 xp + 2xp + 2xp + + + 2 . xk xk+p xk+p xk xk+p xk Since only k is an integration variable, any factor depending on p alone can be pulled out of the integral. Then some of the integrals vanish since they are tadpoles like eq. (1.39). In the following, these integrands are left out entirely. Strictly speaking, they are not zero as functions but only evaluate to zero upon integration due to the vanishing of tadpoles. Note that nontheless we are not dealing with renormalized integrands since the latter would require an explicit subtraction at fixed external momentum. For the 1-loop integrals in this example, kinematic renormalization works even if ρ is non- renormalizable. But we choose to not renormalize the 1-loop case since this would support the wrong impression that renormalization of ρ generally is possible. 1 2 1 1 I(Γ1) = λ 2 xk xk+p 1 1 I(Γ2) = −2λa1xp xk xk+p 2 2 1 1 xk xk+p I(Γ3) = 2a1 xp + + . xk xk+p xk+p xk x The integrands xk and k+p are equivalent upon a shift of the integration variable. xk+p xk Expanding xk+p according to eq. (1.4) produces four integrals and the total result 58 vanishes upon renormalization according to eq. (1.39): Z 4 Z 4 2 2 2 ! d k xk+p d k k + 2kp + p − m 2 = 2 (2π) xk (2π) xk Z 4 Z 4 2 2 2 d k 1 d k k = k + p − m 2 +2p 2 (2π) xk (2π) xk | {z } | {z } →0, renormalization =0, symmetry Consequently, all non-vanishing contributions to the Feynman integrands share a com- mon denominator xkxk+p, 1 2 1 1 I(Γ1) = λ 2 xk xk+p 1 1 I(Γ2) = −2λa1xp (3.18) xk xk+p 2 2 1 1 I(Γ3) = 2a1xp . xk xk+p This reflects the fact that in fig. 3.6 the only non-tadpole graphs have precisely these two propagators. The generic 1-loop-multiedge integrand will appear frequently in the following, we introduce the shorthand 1 M (1) = up to transformation of the integration momentum k. (3.19) xkxk+p (1) Further we define Γ = Γ1 + Γ2 + Γ3 to be the sum of all 1-loop correction graphs. By linearity of the Feynman rules, eq. (3.18) gives (1) 1 2 (1) (1) 2 2 (1) I Γ = I(Γ1) + I(Γ2) + I(Γ3) = λ M − 2λa1xpM + 2a x M . 2 1 p (1) In order to eliminate the 1-loop correction to the two point function FR Γ , we want to choose the diffeomorphism parameters aj such that the Feynman amplitudes of the four graphs add to zero. Since all share the same topology M (1), the integrands can be added right away, (1) 1 2 2 2 (1) I Γ = λ − 2λa1xp + 2a x · M . (3.20) 2 1 p If this were to be integrated, it would yield a logarithmically divergent integral, and (unless the prefactor is included in the subtraction) a single subtraction would be suf- ficient to render it finite. But this is a coincidence for the 1-loop case only, therefore we refrain from any renormalization attempt. We instead retreat to the stance that if the integrand is identically zero, no renormalization is needed. Consequently, the parenthesis has to vanish, 2 λ λ 2 ! 2 − a1 + a1 = 0. 4xp xp 59 This quadratic equation has precisely one solution for a1, s λ λ 2 λ2 a1 = + ± − 2 2xp 2xp 4xp λ = . (3.21) 2xp The uniqueness of this solution comes as no surprise: Since Feynman rules for the full graph are a product of the Feynman rules of the individual constituents, the four combinations of “φ3-vertex” and “diffeomorphism vertex” in fig. 3.6 represent the term 2 φ3-vertex + diffeomorphism vertex with an unique solution. Actually, it is possible to recover the solution eq. (3.21) from the requirement that the diffeomorphism 3-vertex cancels the φ3-vertex: ! λ 2ia1xp + (−iλ) = 0 ⇒ a1 = . (3.22) 2xp The critical observation, here, is that the two summands of the diffeomorphism vertex which cancel the inner propagators, xk and xk+p, do not contribute. This is due to the vanishing of tadpoles and will be the cornerstone of the all-orders argument in section 3.6 below. 3.5 Explicit calculation for two loops Much like the previous section 3.4, this section again serves as a comprehensive exam- ple for computing diffeomorphism coefficients aj to cancel corrections to the two-point function. The reason this second section is included is that section 3.4 adressed primar- ily the mechanism of renormalization but did only include one single graph topology, namely the one in fig. 3.6. For higher loop orders, many different topologies contribute to the total integrand I Γ(l). Therefore one might assume several topologies remain in the end result and a factorization like eq. (3.20) does not work any longer. However, it does work and this section is the simplest example of the non-trivial mechanism which will be exploited in general in section 3.6. Using eq. (2.15) and the result eq. (3.21), the 3-valent diffeomorphism vertex has the Feynman rule iλ iv3 = (x1 + x2 + x3), xp the 4-valent diffeomorphism vertex from eq. (2.15) is 2 iv4 = if4 · (x1 + x2 + x3 + x4) + ig4m 2 2 λ λ 2 = i 6a2 + 2 (x1 + x2 + x3 + x4) + i 2 m . (3.23) xp xp 60 Also, there is the 4-valent interaction vertex eq. (3.3) λ2 −iw4 = −iλh4 = −6i . (3.24) xp There are four different topologies of 1-loop Feynman graphs with two external edges and no tadpoles. We will discuss them one after another (in an arbitrary order) in the following paragraphs. 3.5.1 Topology A Just as the 1-loop computation section 3.4, there also is a correction of multiedge type with 2 loops. This 2-loop multiedge graph is shown in fig. 3.7, it has symmetry factor 1 6 since all three internal edges can be permuted. l l ΓA,1 = ΓA,4 = p l − k −p p l − k −p k + p k + p l l ΓA,2 = ΓA,3 = p l − k −p p l − k −p k + p k + p Figure 3.7: 2-loop graphs of a diffeomorphism of φ3-theory, topology A. The first Feynman integrand is 1 λ2 λ2 i3 I(ΓA,1) = −6i −6i 6 xp xp xlxl−kxk+p 4 λ (2) = 6i 2 M . xp Here, in analogy to eq. (3.19), we introduced 1 M (2) := up to transformation of the integration momenta l, k. (3.25) xlxl−kxk+p This is the integrand of a 2-loop multiedge (i.e. topology A) ignoring vertex Feynman rules and prefactors. As in eq. (3.17),ΓA,2 and ΓA,3 can be combined I(ΓA,2) + I(ΓA,3) 2 2 2 3 1 λ λ λ 2 i = 2 · −6i i 6a2 + 2 (xp + xl + xl−k + xk+p) + i 2 m . 6 xp xp xp xlxl−kxk+p 61 Any cancellation of an inner propagator gives rise to an integrand vanishing upon renormalization due to eq. (1.40) and is left out. The remaining integrands are 2 2 2 λ λ λ 2 1 I(ΓA,2) + I(ΓA,3) = −2i 6a2 + 2 xp + 2 m xp xp xp xlxl−kxk+p 4 2 λ 2 (2) = −12iλ a2 − 2i 3 xp + m M . xp For the last graph only terms cancelling no internal edge survive, 2 2 2 2 3 1 λ λ 2 λ λ 2 i I(ΓA,4) = i 6a2 + 2 xp + i 2 m i 6a2 + 2 xp + i 2 m 6 xp xp xp xp xlxl−kxk+p 2 4 2 2 2 λ 2 1 λ 2 2 4 (2) = 6ia2xp + 2ia2λ + 2ia2 m + i 4 xp + 2xpm + m M . xp 6 xp In total, topology A gives rise to the integrand I(ΓA) = I(ΓA,1) + I(ΓA,2) + I(ΓA,3) + I(ΓA,4) 4 4 25 λ (2) 5 λ 2 (2) 2 (2) = i 2 M − i 3 m M − 10iλ a2M 6 xp 3 xp 2 4 2 2 (2) λ 2 (2) 1 λ 4 (2) + 6ia2xpM + 2ia2 m M + i 4 m M . (3.26) xp 6 xp 3.5.2 Topology B 1 Topology B as shown in fig. 3.8 has the symmetry factor 2 arising from the inner multiedge. In eq. (3.22) we have seen that the diffeomorphism-3-vertex precisely cancels the corresponding φ3-vertex if they lie adjacent to the outer edge with offshell variable xp. Therefore, all Feynman integrals where an outer (i.e. connected to the external edges) φ3-vertex appears will vanish in the sum over all graphs. Consequently, we can safely ignore these graphs and also skip the corresponding terms where an outer diffeomorphism-3-vertex cancels the outer edge. This does not mean that there will be no outer diffeomorphism-3-vertex: The terms where it cancels an internal edge will generally give contributions. 1 iλ iλ i5 I(ΓB,1) = (xk + xk+p)(−iλ)(−iλ) (xk + xk+p) 2 xp xp xkxk+pxkxlxl−k iλ4 1 1 = 2 (xk + xk+p) + 2xp xk+pxkxlxl−k xkxkxlxl−k 4 iλ 1 1 1 xk+p = 2 + + + . 2xp xk+pxlxl−k xkxlxl−k xkxlxl−k xkxkxlxl−k 62 l − k l − k k k k k ΓB,1 = l ΓB,2 = l p −p p −p k + p k + p l − k l − k k k k k ΓB,3 = l ΓB,4 = l p −p p −p k + p k + p Figure 3.8: 2-loop graphs of a diffeomorphism of φ3-theory, topology B. The graphs with an external 3-vertex of interaction type cancel and are not included here. The first summand is the multiedge M (2) from eq. (3.25). The second and third one are tadpoles and vanish. The last one also vanishes since the numerator consists only of summands where p can be factored out of the integral. 4 λ (2) I(ΓB,1) = i 2 M . 2xp 1 iλ 3 i5 I(ΓB,2) = (−iλ)(xk + xk+p)(xk + xl + xl−k)(xk + xk+p) 2 xp xkxk+pxkxlxl−k 4 λ xk 1 1 1 1 1 = −i 3 + + + + + 2xp xk+pxlxl−k xlxl−k xk+pxl−k xkxl−k xk+pxl xkxl 4 λ 1 xk+p 1 xk+p 1 xk+p − i 2 + + + + + . 2xp xlxl−k xkxlxl−k xkxl−k xkxkxl−k xkxl xkxkxl Any summand with two factors in the denominator vanishes due to eq. (1.40) 4 λ xk xk+p xk+p xk+p I(ΓB,2) = −i 3 + + + (3.27) 2xp xlxl−kxk+p xlxl−kxk xkxkxl−k xkxkxl In the third and fourth summand the l-integration factorizes (after shifting l − k → l), hence these terms are products including a tadpole and vanish. The second summand also represents a tadpole since its numerator k2 + 2kp + p2 − m2 contains only p- dependence which can be pulled out of the integral. Just as in eq. (3.17), due to symmetry it is 4 λ xk I(ΓB,3) = I(ΓB,2) = −i 3 . 2xp xlxl−kxk+p 63 Expanding the products yields for the fourth graph of fig. 3.8 λ4 1 I(ΓB,4) = i 4 (xk + xl + xl−k)(xk + xl + xl−k)(xk + xk+p)(xk + xk+p) 2xp xkxk+pxkxlxl−k 4 λ xkxk xk xk+p xk xk+p xk xk+p = i 4 + + + + + + 2xp xk+pxlxl−k xlxl−k xlxl−k xk+pxl−k xkxl−k xk+pxl xkxl 4 λ xk xk+p xl xl xl xk+pxl xk+p + i 4 + + + + + + 2xp xk+pxl−k xkxl−k xk+pxl−k xkxl−k xkxl−k xkxkxl−k xkxk 4 λ xk xk+p xk+p xl−k xl−k xl−k xk+pxl−k + i 4 + + + + + + . 2xp xk+pxl xkxl xkxk xk+pxl xkxl xkxl xkxkxl We exclude the different types of vanishing terms as discussed in the previous steps, all that remains is 4 λ xkxk xk+p(xl−k + xl+k) I(ΓB,4) = i 4 + . (3.28) 2xp xlxl−kxk+p xkxkxl But since 2 2 2 2 2 2 xl−k + xl+k = l − 2lk + k − m + l + 2lk + k − m 2 = 2xl + 2xk + 2m , the second term in eq. (3.28) also vanishes and 4 λ xkxk I(ΓB,4) = i 4 . 2xp xlxl−kxk+p Topology B totally amounts to the integrand 4 4 4 iλ (2) iλ xk iλ xkxk I(ΓB) = 2 M − 3 + 4 . (3.29) 2xp xp xlxl−kxk+p 2xp xlxl−kxk+p 3.5.3 Topology C Figure 3.9 depicts topology C. Again, we do not include graphs with an external 1 interaction vertex. All graphs of topology C have symmetry factor 2 due to the inter- changeability of their internal vertices. This is true even if one of the vertices is φ3 and the other one is diffeomorphism and can be seen if one combines both their Feynman rules to a single type of vertex. 64 k l k l ΓC,1 = k − l ΓC,2 = k − l p −p p −p k + p l + p k + p l + p k l k l ΓC,3 k − l ΓC,4 = k − l p −p p −p k + p l + p k + p l + p Figure 3.9: 2-loop graphs of a diffeomorphism of φ3-theory, topology C. The graphs where an outer vertex is of φ3-type are excluded since they cancel. Up to tadpoles of types which were encountered earlier, the first integrand is 5 1 iλ iλ 2 i I(ΓC,1) = (xk + xk+p) (xl + xl+p)(−iλ) 2 xp xp xkxk−lxk+pxlxl+p 4 1 (2) = iλ 2 M xp where M (2) again is defined in eq. (3.25). The second integrand is 1 iλ 3 i I(ΓC,2) = (−iλ)(xk + xk+p)(xk + xl + xk−l)(xl + xl+p) 2 xp xkxk−lxk+pxlxl+p 4 −iλ xk xk xl xl = 3 + + + . 2xp xk−lxk+pxl+p xk−lxk+pxl xk−lxk+pxl+p xkxk−lxl+p By means of a linear transformation of the integration variables we can obtain the same denominators, 4 −iλ xk+p xk xl+p xl+p I(ΓC,2) = 3 + + + . 2xp xlxl−kxk xk−lxk+pxl xlxl−kxk xk+pxl−kxl The first and third summand factorize into a p-independent integral and vanish upon renormalization as in eq. (3.27). It only remains 4 −iλ xk xl+p I(ΓC,2) = 3 + . 2xp xk−lxk+pxl xk+pxl−kxl It will be helpful later to replace k → l − p − k in the first term. This reproduces the denominator and only changes the numerator to xl−p−k. 65 Since ΓC,2 and ΓC,3 in fig. 3.9 only differ in the assignment of internal momenta, their integrands must be equal up to linear transformation of integration variables and I(ΓC,2) + I(ΓC,3) = 2I(ΓC,2) 4 −iλ xl−p−k + xl+p = 3 . (3.30) xp xk−lxk+pxl After expanding and leaving out tadpoles, the fourth graph in fig. 3.9 has the Feynman integrand 4 iλ xkxl+p xlxk+p I(ΓC,4) = 4 + . (3.31) 2xp xk−lxk+pxl xkxk−lxl+p Transforming k → k −l and l → k in the second summand reproduces the denominator, but changes the numerator such that 4 iλ xkxl+p xkxl−p−k I(ΓC,4) = 4 + . 2xp xk−lxk+pxl xk−lxlxk+p Topology C, in total, has the integrand 4 4 4 iλ (2) iλ xl−p−k + xl+p iλ xk(xl+p + xl−p−k) I(ΓC ) = 2 M − 3 + 4 . (3.32) xp xp xk−lxk+pxl 2xp xk−lxk+pxl Graphically, both the second and the third fraction arise graphically from double- cancelled internal edges. This can be seen from the fact that there is an offshell-variable in the denominator which also appears as a propagator in the original graphs. Therefore two cancellations took place, the first one eliminated the propagator and the second one produced the factor in the numerator. From the discussion in section 2.3 one can expect these double-cancelled offshell variables to decompose into offshell variables of the adjacent edges. Indeed, as shown in fig. 3.10, the two summands in fig. 3.10 do belong to two different permutations of the external momenta of a single double- cancelled edge. As can be seen from the lowest row in fig. 3.10, the third possible permutation of external momenta corresponds to a graph of topology B. In eq. (3.29) we find the summands 4 4 iλ xk iλ xkxk − 3 + 4 . (3.33) xp xlxl−kxk+p 2xp xlxl−kxk+p which are needed to complete a sum according to lemma A.1, 2 xk + xl−p−k + xl+p = xk−l + xk+p + xl + xp + m . (3.34) Now with eq. (3.29) and eq. (3.32) 4 4 4 3 iλ (2) iλ xl−p−k + xl+p + xk iλ xk(xl+p + xl−p−k + xk) I(ΓB) + I(ΓC ) = 2 M − 3 + 4 . 2 xp xp xk−lxk+pxl 2xp xk−lxk+pxl (3.35) 66 k l l − k l + p −l k − l k + p −p p −p k + p l + p k l − k l − k l − p − k −l l −p k + p p −p k + p l − p − k l − k k −k l − k −k k + p p −l −l −p −p k + p Figure 3.10: Three different permutations of the four external momenta (l − k), (+k + p), (−p) and (−l). Shown on the left is the internal edge the permutations give rise to, on the right the corresponding Feynman graph where this edge appears. Inserting eq. (3.34) produces 2 xl−p−k + xl+p + xk 1 1 1 xp + m = + + + (3.36) xk−lxk+pxl xk+pxl xk−lxl xk−lxk+p xk−lxk+pxl where all summands but the last are tadpoles. The last one has the denominator M (2) from eq. (3.25). Similarly 2 xk(xl+p + xk−p−l + xk) x x x xp x x m = k + k + k + k + k , xk−lxk+pxl xk+pxl xk−lxl xk−lxk+pxl xk−lxk+p xk−lxk+pxl where again three summands vanish and we get 4 4 4 3 (2) iλ 2 (2) iλ 2 xk I(ΓB) + I(ΓC ) = iλ 2 M − 3 xp + m M + 4 xp + m . 2xp xp 2xp xk−lxk+pxl (3.37) The transformation k → l − p − k changes x x k → l−p−k , xk−lxlxk+p xk+pxlxl but with k + p = −u and l + p = −s the same term gets the form x x x k → u+p → l+p . xk−lxlxk+p xu−sxs+pxu xl−kxk+pxl 67 These three forms of the same integrand allow to apply eq. (3.34) and we obtain eq. (3.36). Hence, if the above fractions appear as an integrand without any more factors involving the integration variable, one may replace xk 1 2 (2) → (xp + m )M . (3.38) xlxl−kxk+p 3 Note that this procedure is fully consistent with the way we treated the second summand in eq. (3.35): We used all three different forms of the fraction, summed them up and applied eq. (3.36). But we could as well argue that all three summands are actually the same as we did here and then use eq. (3.38). That the three forms are the same can be seen graphically: If in fig. 3.10 the red edge is cancelled, the resulting graph always has topology D as shown in fig. 3.11, no matter if the original graph was topology B or C. Applying eq. (3.38) to eq. (3.37) finally yields 4 4 4 3 (2) iλ 2 (2) iλ 2 2 (2) I(ΓB) + I(ΓC ) = iλ 2 M − 3 xp + m M + 4 xp + m xp + m M 2xp xp 6xp 4 4 4 2 iλ (2) 2 iλ 2 (2) 1 iλ 4 (2) = 2 M − 3 m M + 4 m M . (3.39) 3 xp 3 xp 6 xp 3.5.4 Topology D 1 Topology D, as shown in fig. 3.11, has symmetry factor 2 . The outer interaction 3- vertex can be excluded from the discussion as argued in the previous topologies. The 4-valent diffeomorphism vertex eq. (3.23) has the Feynman rule 2 2 ! 2 λ xp λ 2 iv4 = i 2 6a2 2 + 1 (x1 + x2 + x3 + x4) + i 2 m . xp λ xp The second possible 4-valent vertex is the interaction vertex eq. (3.3) with Feynman rule λ2 −iw4 = −iλh4 = −6i . xp The first integrand is 2 2 ! ! 1 λ a2xp 2 I(ΓD,1) = i 2 1 + 6 2 (xp + xl + xk−l + xk+p) + m 2 xp λ iλ i4 · (−iλ) (xk + xk+p) xp xkxk−lxlxk+p 4 2 ! λ a2xp 1 1 = i 3 1 + 6 2 xp + 2xp λ xk−lxlxk+p xkxk−lxl 1 1 1 1 1 xk+p + + + + + + xk−lxk+p xkxk−l xlxk+p xkxl xk−lxl xkxk−lxl 4 λ 2 1 1 + i 3 m + . 2xp xk−lxlxk+p xkxk−lxl 68 l k l k ΓD,1 = ΓD,2 = p k − l −p p k − l −p k + p k + p k l k l ΓD,3 = ΓD,4 = p k − l −p p k − l −p k + p k + p l k l k ΓD,5 = ΓD,6 = p k − l −p p k − l −p k + p k + p k l k l ΓD,7 = ΓD,8 = p k − l −p p k − l −p k + p k + p Figure 3.11: 2-loop graphs of a diffeomorphism of φ3-theory, topology D. The graphs where an outer vertex is of −iλ-type are excluded since they cancel against the 3-valent diffeomorphism vertex. 69 Leaving out tadpoles and identifying the multiedge integrand eq. (3.25) yields 4 2 ! 4 λ a2xp (2) λ 2 (2) I(ΓD,1) = i 2 1 + 6 2 M + i 3 m M . 2xp λ 2xp All graphs in fig. 3.11 appear in pairs such that I(ΓD,3) = I(ΓD,1) I(ΓD,4) = I(ΓD,2) I(ΓD,7) = I(ΓD,5) I(ΓD,8) = I(ΓD,6). The second graph of fig. 3.11 produces the integrand 4 2 ! ! λ a2xp 2 I(ΓD,2) = −i 4 1 + 6 2 (xp + xl + xk−l + xk+p) + m 2xp λ 1 1 · (xl + xk + xk−l) + xk−lxlxk+p xkxk−lxl 4 2 ! λ a2xp 1 1 xk 1 1 1 = −i 4 1 + 6 2 xp + + + + + 2xp λ xk−lxk+p xkxk−l xk−lxlxk+p xk−lxl xlxk+p xkxl 4 2 ! λ a2xp xl xl xk 1 1 1 − i 4 1 + 6 2 + + + + + 2xp λ xk−lxk+p xkxk−l xk−lxk+p xk−l xk+p xk 4 2 ! λ a2xp 1 1 xk 1 xk−l xk−l − i 4 1 + 6 2 + + + + + 2xp λ xk+p xk xlxk+p xl xlxk+p xkxl 4 2 ! λ a2xp 1 xk+p xk xk+p 1 xk+p − i 4 1 + 6 2 + + + + + 2xp λ xk−l xkxk−l xk−lxl xk−lxl xl xkxl 4 λ 2 1 1 xk 1 1 1 − i 4 m + + + + + , 2xp xk−lxk+p xkxk−l xk−lxlxk+p xk−lxl xlxk+p xkxl but after excluding all tadpoles, what remains is 4 2 ! 4 λ a2xp xk λ 2 xk I(ΓD,2) = −i 4 1 + 6 2 xp − i 4 m . 2xp λ xk−lxlxk+p 2xp xk−lxlxk+p We use eq. (3.38) to obtain 4 2 ! 4 λ a2xp 2 (2) λ 2 2 (2) I(ΓD,2) = −i 3 1 + 6 2 (xp + m )M − i 4 m (xp + m )M . 6xp λ 6xp The remaining graphs in fig. 3.11 contain an interaction 4-vertex, which has a much 70 easier Feynman rule eq. (3.24). 1 λ2 iλ i4 I(ΓD,5) = −6i (−iλ) (xk + xk+p) 2 xp xp xkxk−lxlxk+p λ4 1 1 = −3i 2 + xp xk−lxlxk+p xkxk−lxl 4 λ (2) = −3i 2 M . xp 1 λ2 iλ iλ i4 i4 I(ΓD,6) = −6i (xl + xk + xk−l) + 2 xp xp xp xk−lxlxk+p xkxk−lxl 4 λ 1 1 xk 1 1 1 = 3i 3 + + + + + . xp xk−lxk+p xkxk−l xk−lxlxk+p xk−lxl xlxk+p xkxl Excluding tadpoles and using eq. (3.38) produces 4 λ 2 (2) I(ΓD,6) = i 3 (xp + m )M . xp Even for Topology D all integrands are proportional to M (2) and I(ΓD) = I(ΓD,1) + 2I(ΓD,2) + 2I(ΓD,5) + 2I(ΓD,6) 4 2 ! 4 λ a2xp (2) λ 2 (2) = i 2 1 + 6 2 M + i 3 m M xp λ xp 4 2 ! 4 λ a2xp 2 (2) λ 2 2 (2) − i 3 1 + 6 2 (xp + m )M − i 4 m (xp + m )M 3xp λ 3xp 4 4 λ (2) λ 2 (2) − 6i 2 M + 2i 3 (xp + m )M xp xp 4 4 2 4 10 λ 2 7 λ 2 λ 2 1 λ 4 (2) = − i 2 + 4iλ a2 + i 3 m − 2i m a2 − i 4 m M . (3.40) 3 xp 3 xp xp 3 xp 71 3.5.5 Total result For the total 2-loop correction to the 2-point function, we have to sum up the contri- butions of all four topologies, namely eqs. (3.26), (3.39) and (3.40). 4 4 (2) 25 iλ (2) 5 iλ 2 (2) 2 (2) 2 2 (2) I Γ = 2 M − 3 m M − 10iλ a2M + 6ia2xpM 6 xp 3 xp 2 4 iλ 2 (2) 1 iλ 4 (2) + 2a2 m M + 4 m M xp 6 xp 4 4 4 4 2 iλ (2) 2 iλ 2 (2) 1 iλ 4 (2) 10 iλ (2) + 2 M − 3 m M + 4 m M − 2 M 3 xp 3 xp 6 xp 3 xp 4 2 4 2 (2) 7 iλ 2 (2) iλ 2 (2) 1 iλ 4 (2) + 4iλ a2M + 3 m M − 2 m a2M − 4 m M 3 xp xp 3 xp 4 1 λ 2 2 2 (2) = i 9 2 − 36λ a2 + 36a2xp M . (3.41) 6 xp Althought four different topologies contributed and lots of intermediate terms appeared, the overall integrand turns out to contain only a single topology. Hence the Feynman amplitude factorizes just as in eq. (3.20). The integral over M (2) is quadratically di- vergent and can be renormalized in principle, but instead of considering a renormalized amplitude, we require 4 (2) 1 λ 2 2 2 (2) ! I Γ = i 9 2 − 36λ a2 + 36a2xp M = 0. (3.42) 6 xp The prefactor has to vanish and we obtain for the diffeomorphism parameter the quadratic equation 4 2 λ λ 2 4 + 3 2 a2 + a2 = 0. 4xp xp It has an unique solution, λ2 a2 = 2 . (3.43) 2xp This result is surprisingly simple and motivates the assumption that even in higher loop orders no topologies apart from multiedges remain. If that is the case, the dif- feomorphism parameters factorize and can be read off without solving any Feynman integral. 3.6 All-orders cancellation of corrections The procedure we use to determine arbitrary higher diffeomorphism parameters relies on the special role of higher interaction vertices theorem 3.1 and the structure of interaction tree sums lemma 3.1 as well as the assumption that any tadpole graph vanishes. 72 3.6.1 Decomposition of non-vanishing graphs In both explicit examples eqs. (3.20) and (3.42) the sum of all graphs of a given loop order l involved only integrals of one single topology, namely the multiedge on l loops M (l). This can be understood from the vanishing of tadpoles as was noted in section 2.4: Consider an arbitrary l-loop graph Γ contributing to the 2-point function. Assume further there is an internal 2-edge loop just like the upper loop in fig. 3.8 somewhere in Γ. One sums over all ways of assigning diffeomorphism- or interaction vertices to the vertices of Γ. Hence, there will be summands where the edges of the internal loop are cancelled and summands where this is not the case. Graphically, cancelling the edge amounts to identifying its adjacent vertices, but this will produce a tadpole from the other edge in the loop. Hence for the graph to contribute, both edges of the internal loop must not be cancelled. This argument continues for any loop consisting of more than two edges. For example the lower loop in fig. 3.8 consisting of the four edges k, k + p, k and l. It is possible to cancel both k edges in this case, but at least two edges have to remain uncancelled. In the following it is helpful to consider uncancelled edges instead of cancelled ones. Since for diffeomorphism vertex rules, an edge e being uncancelled is equal to setting this edge onshell (xe = 0, see section 2.3), this resembles the notion of “cut” edges from Cutkosky’s theorem. Note however that this does not mean the edge is onshell in the Feynman integral, it will still be integrated over and there still is an propagator i . xe But from the point of view of diffeomorphism vertices, one can consider such an edge onshell and speak of “cutting” the graph. Applying diffeomorphism Feynman rules, all uncut edges shrink to zero length and only the cut edges survive. To avoid tadpole graphs, all loops must be cut, i.e. there must not be any closed path of uncut lines in the graph. Additionally, there must not be a path of uncut lines between the two external vertices. Having such a path implies identifying the two external vertices and renders the whole graph a tadpole. Lemma 3.3 Decomposition. Let Γ be a 2-connected graph with two external vertices (i.e. a 1PI-contribution to the 2-point function) which, upon applying edge cancellation due to the vertex Feynman rules eq. (2.15), is no tadpole. If one takes the uncancelled edges to be cut, i.e. to divide the graph, then Γ decomposes into connected components which are trees and the two external vertices of Γ lie in different connected components. Proof. Follows from the above discussion. Requesting Γ to be 2-connected means it is not itself a tree, consequently both external vertices belong to (one or more) loops. To be no tadpole, there needs to be an overall momentum flow through Γ, this means the external vertices must not be identified. Further, there must not be any uncut path between the external vertices, hence they lie in different connected components. If any uncut closed path remains, it will become a tadpole upon contraction of uncut edges. Since the connected components contain no closed path, they are trees. 73 3.6.2 Factorization of tree sums By the decomposition lemma (3.3) the integrand of any fixed graph Γ is a product of trees. For the total correction to the 2-point function, one has to sum over all such Γ. This can be done by fixing all except for one tree-component of Γ and summing over all possible rearrangements and permutations of the remaining tree. Since all graphs have to be produced, this is a sum over all trees with a given number of external onshell edges and (possibly) one external offshell edge. It is not trivial to see if this procedure really reproduces all graphs Γ with their correct symmetry factor. But by lemma A.3 it actually does: Connecting trees in all possible ways produces just the right symmetry factors up to mul(Γ,C), i.e. the number of equivalent Cutkosky cuts. Indeed, this is what is needed here: Consider ΓC,4 from fig. 3.9 which admits two equivalent Cutkosky cuts. Seemingly, in the sum over all connected trees, this graph will appear twice as often as it should. But the Feynman rules of the diffeomorphism vertex actually produce two terms: One in which k, k − l and l + p are uncancelled and one in which l, k − l and l + p are uncancelled, see eq. (3.31). These two terms directly correspond to the two different Cutkosky cuts ΓA and ΓB in example 22. This phenomenon works even for bigger graphs: Admitting two equivalent Cutkosky cuts and admitting two different ways of (non-vanishing) cancellation of inner edges is the same thing. Note that for bigger graphs contributing to the 2-point function there will be no decomposition into just two trees. But this is not necessary: By lemma 3.3 there is a decomposition into trees, so it is always safe to assume each of the external vertices belongs to a tree. Possibly there are more “intermediate” trees. Nontheless lemma A.3 also holds for more than two components. One can even choose one of the external vertices, call it the left one, and use the corresponding tree as Γ1 and the whole rest of the graph as Γ2. Then, Γ2 generally is no tree, but still lemma A.3 can be applied. Even if the graph does not decompose into two tree sums, it is still proportional to one tre esum. Lemma 3.4 Factorization of interaction tree sums. For a fixed loop number l, and if Γ(l) is the sum of all graphs contributing to the 2- point function, then its non-tadpole Feynman integrand I Γ(l) is a sum of terms each 0 of which is proportional to one of the tree sums bn for n ≤ l + 1. Proof. By lemma 3.3 and the above discussion it is always possible to split off a tree on the left side of the graph and in the sum over all graphs this tree will resemble 0 the sum over all trees, i.e. bn. Using the Euler characteristic section 1.5.2, for a l- loop graph, there are never more than l + 1 edges to be cut to decompose the graph 0 into two connected components, hence bl+1 is the biggest tree sum which can possibly appear. Note that lemma 3.4 does not yet mean that the Feynman integral F Γ(l) factorizes 0 into some tree sum multiplied by an integral. Generically, since bn involve kinematic variables, it is not possible to exclude them from the integration. 74 Example 11: Decomposition of a 3-loop graph As an example for lemma 3.4, take Γ1 to be the trees from b3 and Γ2 a graph with one internal loop. Γ1 = + Γ2 = 1 2 Let further Γ1 be the left and Γ1 the right connected component of Γ1. Then 1 sym(Γ1) = 2 2 sym(Γ1) = 6 sym(Γ2) = 4, the latter due to the exchange of the two leftmost external legs as well as the exchange of the two edges in the internal multiedge. The possible outcomes of connecting Γ1 and Γ2 are shown below. l m − k k k k Γ = m A p l − k −p k + p m − k l k k Γ = l − k m B p −p l + p k + p m − k l k k Γ = m C p l − k −p k + p Since there are three edges to be connected, there are 3! = 6 possible ways of 75 assigning them to each other. With the definitions from lemma A.3, 1 F Γ1, Γ2 = 2ΓA + 4ΓB 2 F Γ1, Γ2 = 6ΓC . The correct resulting symmetry factors are 1 1 1 sym(ΓA) = sym(ΓB) = sym(ΓC ) = . 4 2 4 Using the procedure from lemma A.3 indeed reproduces them: 1 1 1 1 F Γ1, Γ2 = (2ΓA + 4ΓB) (3.44) sym(Γ1) sym(Γ2) 2 · 4 1 1 1 2 F Γ1, Γ2 = 6ΓC , sym(Γ1) sym(Γ2) 6 · 4 so 1 1 1 1 1 F Γ1, Γ2 + 2 F Γ1, Γ2 (3.45) sym(Γ1) sym(Γ2) sym(Γ1) sym(Γ2) 1 1 1 = ΓA + ΓB + ΓC . (3.46) 4 2 4 The procedure however cannot be directly applied to sums of subgraphs in a naive 1 2 way such as F Γ1 + Γ1, Γ2 because each individual summand needs to be mul- tiplied by its corresponding inverse symmetry factor. The problem can be cured if one uses the graphs including their proper symmetry factors, i.e. 1 2 Γ1 Γ1 Γ2 F 1 + 2 , sym(Γ1) sym(Γ1) sym(Γ2) will obviously reproduce the correct result eq. (3.45). But this also is what actually happens if one uses b3 instead of Γ1: As stated explicitly in example 4, b3 consists of 2 1 Γ1 and three different versions of Γ1 where the unique external leg is assigned one of the three external momenta, respectively. But the operator F from lemma A.3 always sums over all possible ways of connecting edges, hence each of these three graphs in b3 must necessarily give the same result if F acts on them. Explicitly, 1 2 F ({b3, Γ2}) = F 3Γ1 + Γ1, Γ2 1 2 = 3F Γ1, Γ2 + F Γ1, Γ2 . 1 The relative factor of 3 between the summands just reproduces the prefactors 2·4 1 and 6·4 in eq. (3.44) up to an overall factor. This phenomenon is no random coincidence for b3 but actually follows from the orbit-stabilizer-theorem: The operator F sums over just all ways to connect edges, which is the symmetric group Sn for n edges with n! permutations. On the other hand the symmetry factor sym(Γ) is the number of permutations of the n edges 76 which do not alter a planar version of the graph, i.e. exchanges of equivalent edges in the usual sense. Then, summing over Sn and dividing by sym(Γ) as in lemma A.3 is the same as multiplying with the number of permutations which exchange non- equivalent edges. But in bn, it is precisely these permutations of non-equivalent edges which the graphs are summed over. Hence using bn instead of the naive sum of trees (where each tree would be weighted equally) will always reproduce 0 the symmetry factors up to an overall constant factor. The same is true for bn instead of bn since the combinatorics there is just the same. Therefore lemma 3.4 is true even if the operation performed there is slightly different from the one in lemma A.3. Lemma 3.5 Factorization of integration. For a fixed loop number l > 1, if the diffeomorphism parameters aj have been chosen such that all contributions Γ(s) to the 2-point function with loopnumber s < l vanish, then the remaining non-tadpole Feynman integrand of Γ(l) reduces to 3+l i 2 I Γ(l) = x2 · b · M (l), (l + 1)! p l+1 where M (l) is the Feynman Integrand of a l-loop multiedge with vertex amplitudes 1 and external momentum p. Proof. The statement holds for 1 and 2 loops by the examples sections 3.4 and 3.5, see eqs. (3.20) and (3.42). Assume it holds for l −1 loops. This means, aj for j ≤ l −1 have 0 been chosen such that bk = 0 for all k ≤ l. By lemma 3.2 this also implies bk = 0∀k ≤ l. Then by lemma 3.4, the only remaining part of the Feynman integrand of the l-loop 0 graphs is proportional to bl+1. Since all lower bk are already zero, by lemma 3.1 it is actually proportional to the pure diffeomorphism tree sum bl+1. But when the edge cancellation imposed by the diffeomorphism vertex Feynman rules is carried out, any pure diffeomorphism tree sum bk reduces to a (k + 1)-valent vertex. Hence the graphs (l+1) in Γ which are proportional to bl+1 must, after cancellation, have a topology where there is a (l + 2)-valent vertex at at least one side. The only l-loop topology without tadpoles where this can happen is the l-loop multiedge with integrand I(i). Hence all remaining Feynman integrals involve the same topology and the integration factors. The l-loop multiedge contains two (l + 2)-valent vertices, hence the prefactor must be 2 1 (l) (bl+1) . The factor (l+1)! is the symmetry factor of the l-loop multiedge. M contains l+1 edges giving (i)l+1 if the factors of i are not included into the integrand M (l). There is no factor i from vertices since viewing bl+1 as a vertex, it actually has Feynman rule bl+1 and not ibl+1 (all factors i have already been included in the computation of bn, see e.g. example 4) Finally, the external propagators were included in bl+1, they must 2 be cancelling by multiplication with (−ixp) . 77 Example 12: 2-loop graphs Consider the 2-loop graphs from section 3.5. There, a1 was chosen in such a way that 1-loop graphs Γ(1) are cancelled, therefore the conditions of lemma 3.5 are fulfilled. Indeed, the explicitly computed result eq. (3.42) is 4 (2) 1 λ 2 2 2 (2) I Γ = i 9 2 − 36λ a2 + 36a2xp M 6 xp 5 4 i 2 λ 2 a2 2 (2) = xp 9 4 − 36λ 2 + 36a2 M . 3! xp xp Using eq. (3.21) λ λ a1 = ⇒ = 2a1, 2xp xp the parenthesis becomes 4 2 2 4 2 2 9(2a1) − 36(2a1) a2 + 36a2 = 144a1 − 144a1a2 + 36a2 2 2 = 12a1 − 6a2 . According to example 4 this is b3, thus 5 (2) i 2 2 (2) I Γ = x (b3) M . 3! p Example 13: Topology of 3-loop graphs Consider again the above example example 11. On first sight it might seem ques- tionable that the three Feynman integrands ΓA, ΓB, ΓC indeed cancel, i.e. that they reduce to some prefactor multiplied by an integral over the same topology. Assume for convenience that no edge in Γ2 is cancelled. But in Γ1, the internal edge in the tree consisting of two vertices will be cancelled: We know b3 does not 2 contain any internal propagator, but is just b3 = 12a1 − 6a2. Consequently, in terms of topology, b3 is a single vertex. After executing the cancellation, all three graphs have the same topology ΓC . This is not a 3-loop multiedge as promised by lemma 3.5 because the order is wrong: The multiedge is produced by connecting b4 with a rest (which, as discussed, has to be b4, too, to produce a 3-loop graph). Conversely, in the conditions of lemma 3.5 the assumption that 2-loop integrals cancel already implies b3 = 0 as can be checked from eqs. (3.21) and (3.43). There- fore, the topology ΓC does not appear in the final result of the 3-loop integrand and lemma 3.5 is not violated. 78 3.6.3 Parameters of the adiabatic diffeomorphism Lemma 3.6. If Cn are the Catalan numbers from eq. (1.5), a1 is arbitrary and n an = Cna1 ∀n ≥ 2, (3.47) then bn = 0 ∀n ≥ 3. n Proof. Note that, since C1 = 1, the formula an = Cna1 even holds for n = 1. By theorem 2.1 and using eq. (1.15) n X (n + k)! 2 n bn+1 = B −1!C1a1, −2!C2a ,..., −n!Cna n! n,k 1 1 k=1 n (n + 1)! n X (n + k)! k n = (−n!Cna ) + (−1) (a1) B (1!C1, 2!C2, . . . , n!Cn) n! 1 n! n,k k=2 The value of the Bell polynomial has been computed in lemma A.13, so n (2n)! n n (2n)! X k (n − 1)! bn+1 = − a + a (−1) . n! 1 1 n! (k − 1)!(n − k)! k=2 The remaining sum equals one by explicit calculation: n n−1 X (n − 1)! X n − 1 (−1)k = − (−1)k (k − 1)!(n − k)! k k=2 k=1 n−1 ! X n − 1 n − 1 = − (−1)k1(n−1)−k − (−1)0 k 0 k=0 = − (1 + (−1))n−1 − 1 = 1. Theorem 3.2. λ 3 Let φ(x) be a scalar quantum field with interaction term − 3! φ according to eq. (3.2) λρ 1 and p a fixed offshell 4-momentum. Assume all tadpole graphs vanish and p2−m2 ≥ − 2 . Then the field formally given by λ ρ(x) = φ(x) − φ2(x) 2(p2 − m2) has a free 2-point function at that very momentum, 2 2 −1 ρ(p)ρ(−p) = −i(p − m + i0) = ∆F (p). 79 Proof. In order to have a free 2-point function, all Feynman graphs with two external edges have to be cancelled. Since they can be connected arbitrarily, it is sufficient to cancel all 2-connected, i.e. 1PI, graphs. The solution for a1 has been obtained in eq. (3.21) by explicit computation. Assume all ak are known for k < l. Then by lemma 3.5 the first non-vanishing term arises from the l-loop integrand which factorizes into some (non-vanishing) integration and a 2 ! prefactor (bl+1) . So the term is cancelled by requiring bl+1 = 0. These cancellations are given by lemma 3.6, so to cancel all corrections to the 2-point function one has to set a0 = 1 λ a1 = 2xp n an = Cna1 ∀n ≥ 1. The diffeomorphism eq. (3.1) takes the form ∞ X n n φ = ρ · Cna1 ρ . (3.48) n=0 This is the generating function of the Catalan numbers, eq. (1.11), with argument a1ρ, namely √ 1 − 1 − 4a1ρ 1 p φ = ρ · = 1 − 1 − 4a1ρ . 2a1ρ 2a1 1 The series is convergent and a monotonous function for a1ρ ≤ 4 and can thus be 2 inverted giving ρ = φ − a1φ . For λ → 0 and constant xp =6 0 one has limλ→0 a1 = 0 and thus lim ρ = φ. λ→0 If the original interaction of the φ3-theory vanishes, then φ itself has a free propagator and the diffeomorphism becomes the identity and both fields coincide as they should. In fig. 3.12, the diffeomorphism φ(ρ) is shown for different values of a = λ . The 1 2xp condition 1 ρ ≤ a1 4 is obvious here, it is the point where the function becomes vertical. The inverse function ρ(φ) from theorem 3.2 is a parabola, so it can be readily extended beyond the interval where it is single-valued as a function φ(ρ). Thus in fig. 3.13 the function is shown for all φ. All parabolas intersect the origin with a slope of one, this is enforced by the condition that the diffeomorphism eq. (3.1) be tangent to identity. 80 Figure 3.12: φ(ρ, a1) for different values of a1 in steps of two and the special case a1 = 0. Figure 3.13: The inverse diffeomorphism ρ(φ) converting the φ3-field φ into a field ρ with free propagator according to theorem 3.2. It is a parabola. Shown are again different values of a1 in steps of two. 81 4 Higher order interaction 4.1 Feynman rules The results from chapter 3 can be readily generalized to an interaction term with higher valence as long as kinematic renormalization can be used implicitly in all the arguments about vanishing tadpoles. Consider the Lagrangian density 1 µ 1 2 2 λ s L(x) = − φ(x)∂µ∂ φ(x) − m φ (x) − φ (x) (4.1) 2 2 s! and apply the same diffeomorphism as eq. (3.1), ∞ X j+1 φ(x) = ajρ (x). (4.2) j=0 By direct analogy to eq. (3.3), there are interaction vertices with valence n ≥ s and Feynman rule n−s n−s−j n−s−j−k−... λ X X X −iwn = −i n! ··· aja ··· a a . s! k l n−s−j−k−...−l j=0 k=0 l=0 | {z } s factors of a | {z } s−1 sums The multi-sum actually gives all s-fold products ak1 ak2 ··· aks such that kj ≥ 0 and s k1 + k2 + ... = n − s. By Fa`adi Bruno‘s formula eq. (1.19), if f(x) = x and g(x) = 2 a0x + a1x + ..., then n s! [x ]f(g(x)) = Bn,s(1!a0, 2!a1, 3!a2, 4!a3,...). n! The prefactor is just cancelled in wn because the sum over all permutations introduces a factor n! and the Lagrangian eq. (4.1) involves (s!)−1, so −iwn = −iλBn,s(1!a0, 2!a1, 3!a2, 4!a3,...). (4.3) 83 Example 14: Vertex Feynman rules of φ4-theory Analogous to section 3.1, in φ4-theory the Lagrangian density 1 µ 1 2 2 λ 4 L = − ∂µ∂ φ − m φ − φ (4.4) φ 2 4! produces an interaction term ∞ n−4 n−4−j n−4−j−k λ 4 λ X X X X n − φ = − aja a a ρ . 4! 4! k l n−4−j−k−l n=4 j=0 k=0 l=0 Now for n ≥ 4 there is a vertex with Feynman rule n−4 n−4−j n−4−j−k λ X X X −iwn = −i n! aja a a 4! k l n−4−j−k−l j=0 k=0 l=0 = −iλhn with n−4 n−4−j n−4−j−k n! X X X hn = aja a a 4! k l n−4−j−k−l j=0 k=0 l=0 = Bn,4(1, 2!a1, 3!a2, 4!a3,...). The first of these parameters are h4 = 1 h5 = 20a1 2 h6 = 120a2 + 180a1, similar to fig. 3.1 the vertices are shown below: −iw4 = = −iλ −iw5 = = −20iλa1 2 −iw6 = = −60iλ 2a2 + 3a1 . . 84 4.2 Tree sums and higher interaction vertices The central ingredient for understanding φ3-diffeomorphisms was theorem 3.1, the state- ment that onshell tree sums vanish. To establish this fact for φs-theory, one can repeat the construction from section 3.2.3: 1 Define Sn to be the sum of all trees proportional to λ , i.e. all trees consisting precisely one interaction vertex, and n external edges. The remaining edges are filled up with pure diffeomorphism trees bki . By the same argument as in section 3.2.3, this amounts to taking the Bell polynomial of the trees bi and n X Sn = − iwkBn,k(b1, b2,...). k=s s Sn resembles eq. (3.6) with just two differences: The wk are interaction vertices of φ - theory according to eq. (4.3) instead of the ones from eq. (3.3) and consequently the sum starts at k = s instead of k = 3. Inserting the vertex Feynman rules eq. (4.3) gives the explicit form n X Sn = −iλ Bn,k(b1, b2,...)Bk,s(1, 2!a1, 3!a2, 4!a3,...). (4.5) k=s Theorem 4.1. If Sn is the onshell tree sum eq. (4.5) of interactions caused by a diffeomorphism of s scalar φ -theory according to eq. (4.1) , then Ss = −iλ and Sn = 0 for any n =6 s. Proof. Sn vanishes for n < s since it is an empty sum. For n = s the result iλ is obvious s from Bn,n(...) = 1. To be shown is Sn = 0 for n > s. Define Sn = −(−1) iλsn to eliminate unnecessary prefactors. The bk depend on ak via theorem 2.1. First apply lemma A.15, n−k X (n − 1 + j)! B (b1, b2,...) = B (−1!a1, −2!a2,...) 0 < k < n. n,k (k − 1)!(n − k)! n−k,j j=0 Since k < n is required, the last summand in eq. (4.5) has to be split off, using 1 = Bn,n(...) this is n−1 s X sn = (−1) Bn,k(b1, b2,...)Bk,s(1, 2!a1, 3!a2,...) k=s s + (−1) Bn,s(1, 2!a1, 3!a2,...) n−1 n−k s X X (n − 1 + j)! = (−1) B (−1!a1, −2!a2,...)B (1, 2!a1, 3!a2,...) (k − 1)!(n − k)! n−k,j k,s k=s j=0 (4.6) s + (−1) Bn,s(1, 2!a1, 3!a2, 4!a3,...). 85 With 0 X (n − 1 + j)! (n − 1)! B0,j(−1!a1, −2!a2,...) = B0,0(−1!a1, −2!a2,...) = 1, (n − 1)!0! (n − 1)! j=0 the last summand k = n can be taken into the sum eq. (3.9), n n−k s X X (n − 1 + j)! sn = (−1) B (−1!a1, −2!a2,...)B (1!a0, 2!a1, 3!a2,...) (k − 1)!(n − k)! n−k,j k,s k=s j=0 (4.7) Substitute −j!aj → xj, which implies (j + 1)!aj → −(j + 1)xj and x0 = −0!a0 = −1. n n−k s X X (n − 1 + j)! sn = (−1) B (x1, x2,...)B (−1x0, −2x1, −3x2,...). (k − 1)!(n − k)! n−k,j k,s k=s j=0 The remaining signs can be extracted from the Bell polynomial, giving an additional (−1)s, i.e. n n−k X X (n − 1 + j)! sn = B (x1, x2,...)B (1x0, 2x1, 3x2,...). (k − 1)!(n − k)! n−k,j k,s k=s j=0 This is zero by lemma A.12 since n =6 s. Since the higher interaction vertices behave similarly as in φ3-theory, the discussion 0 s of interacting tree sums bn from section 3.3 still applies for φ theory. Especially, the s-valent interaction vertex −iws = −iλ is the only one to survive. For s > 3, this means that there will be small trees which are not big enough to fit a single interaction vertex. 0 Therefore bn = bn for all n < s − 1. The straightforward generalization of lemma 3.1 is Lemma 4.1. s 0 In φ -theory according to eq. (4.1), bn consists of pure diffeomorphism tree sums bk for s < k < n which are multiplied by symmetric sums of internal propagators and an arbitrary number of interaction vertices −iws = −iλ. For n = s − 1, the interaction vertex itself appears without any attached diffeomorphism vertices and 0 λ bs−1 = bs−1 + . (4.8) xp For n < s − 1 interaction does not alter the tree sums, 0 bn = bn n < s − 1. Proof. Argument from section 3.3: All higher interaction vertices with uncancelled edges sum to zero by theorem 4.1. The ones with cancelled edges contribute to some sum which in the end still adds up to zero. Hence only s-valent interaction vertices with uncancelled edges remain. In tree sums, uncancelled edges must be adjacent to an 86 interaction vertex or they are the (onshell) lower edges of some bk. If the edges adjacent to the interaction vertex are not cancelled, the vertex is either connected to another interaction vertex or to an bk. In either case, the internal propagator remains and in the sum over all trees, this becomes a symmetric expression. As a direct consequence of lemma 4.1 we once more obtain Lemma 4.2. s 0 In a diffeomorphism of φ -theory with tree sums bn with n external onshell edges and one edge offshell, including all interaction vertices wn: 0 bn vanish for all kinematic configurations 0 ⇔ bk = 0 ∀1 < k ≤ n, k =6 s − 1 and bs−1 = 0. Proof. See proof of lemma 3.2. 4.3 Cancellation of corrections to the 2-point function Lemma 4.3. In φs-theory, a necessary condition for the cancellation of corrections to the 2-point function at a given momentum p is aj = 0 ∀j ∈ {1, . . . , s − 3} λ as−2 = . (s − 1)!xp Proof. The lemmas 3.3 to 3.5 are graph-theoretical and do not depend on the underlying theory. They hold even for φs- theory. Consequently, to cancel the corrections to the 0 2-point function, we have to impose bn = 0 for all n. By lemma 4.2 we first need bk = 0 ∀1 < k < s − 1. Since by theorem 2.1 the bk are a homogeneous polynomial of aj with 1 ≤ j < k, this can be obtained by setting aj = 0 ∀1 ≤ j < s − 2. (4.9) Under the condition eq. (4.9), by theorem 2.1 and eq. (1.15) only the summand k = 1 remains for bs−1, namely s−2 X (s − 2 + k)! bs−1 = B (0, 0,..., 0, −(s − 2)!as−2) (s − 2)! s−2,k k=1 (s − 2 + 1)! = (−(s − 2)!as−2) (s − 2)! = −(s − 1)!as−2. 87 Then the requirement eq. (4.8) from lemma 4.2, λ bs−1 = − , xp fixes as−2 as stated. Example 15: First coefficient for φ4-theory After setting a1 = 0, the only remaining non-tadpoles at two loops are the multi- edges k + l + p k Γ = 1 p −p l k + l + p k + l + p k k Γ = Γ = 2 p −p 3 p −p l l k + l + p k Γ = 4 p −p l 1 They have symmetry factor 6 and explicitly computing their graphs yields 1 2 i i i I(Γ1) = (−iλ) 6 xk xl xk+l+p 1 = iλ2M (2), 6 where M (2) is the prototypic integrand of the 2-loop multiedge as in eq. (3.25). By eq. (2.15), the pure diffeomorphism 4-vertex with a1 = 0 is iv4 = 6ia2(x1 + x2 + x3 + x4). Just like in eq. (3.17), the symmetric pair of graphs gives rise to two times the same integrand, (2) I(Γ2 + Γ3) = −2iλa2xpM . 88 Finally 2 2 (2) I(Γ4) = 6ia2xpM . For the 2-point function to vanish, ! 0 = I(Γ1 + Γ2 + Γ3 + Γ4) 1 2 2 2 (2) = iλ − 2iλa2xp + 6ia x M . 6 2 p The condition 2 2 ! λ − 12λa2xp + 36(a2xp) = 0 2 with b3 = −6a2 + 12a1 = −6a2 actually is just the square of eq. (4.8) and thus its solution λ a2 = 6xp reproduces lemma 4.3. The remaining diffeomorphism coefficients aj for j ≥ s − 1 have to be fixed recursively by bn = 0 for n > s − 1. This condition implies a unique solution. The smallest non-vanishing vertex in tree sums has valence s, therefore only trees which allow for an integer multiple of these vertices are possible. Such trees can have s, 2s − 2, 2s − 2 + (s − 2),... external edges. Let the number of vertices be v > 1, then there are y = 2(s − 1) + (v − 2)(s − 2) external edges. Conversely, all other tree sums bn for n =6 y − 1 can be cancelled by just setting aj = 0 for j < s − 2 (which has already been done in lemma 4.3). But the tree sum bn fixes the coefficient an−1, hence aj = 0 j =6 y − 2. If we put in the value of y, we find y − 2 = 2s − 4 + (v − 2)(s − 2) = v(s − 2). This expression is valid even for v = 1, namely there a single vertex has s edges, giving rise to bs−1 fixing as−2. Next, two vertices have 2s − 2 external legs, fixing a2s−4 = a2(s−2). What remains is to determine the coefficients a2(s−2). To do this, we need a general- ization of lemma 3.6: 89 Lemma 4.4. For an integer s ≥ 3, if cn = An(s − 1, 1) are the Fuss-Catalan numbers definition 2 and bn is from theorem 2.1 and j aj·(s−2) = cjas−2, j ∈ N ak = 0, k∈ / N · (s − 2) then bn = 0 ∀n ≥ s. Proof. Similar to the proof of lemma 3.6. By theorem 2.1, n X (n + k)! bn+1 = B (0,..., 0, −(s − 2)!as−2, 0,...) n! n,k k=1 n X (n + k)! 1 s−3 s−2 2−1 = B 0a s−2 ,..., 0a s−2 , −(s − 2)!a s−2 , 0a s−2 ,... n! n,k s−2 s−2 s−2 s−2 k=1 n 1 n (n + 1)! X (n + k)! k s−2 = (−n!an) + (−1) a B (...). (4.10) n! n! s−2 n,k k=2 Lemma A.14 with the choice m = s − 2 gives the value of the Bell polynomial. Let n = j(s − 2) with integer an j, then n (n − 1)! s−2 (s − 1) Bn,k(...) = (s − 2) n (k − 1)! s−2 − k j and an = aj(s−2) = cjas−2 by assumption. Therefore eq. (4.10) takes the form n n n j X (n + k)! k s−2 1 s−2 (s − 1) bn+1 = −(n + 1)!cja + (s − 2)(−1) a s−2 n s−2 (k − 1)! n − k k=2 s−2 j(s−2) j j X (j(s − 2) + k)! k 1 j(s − 1) = −(j(s − 2) + 1)!cja + a (−1) s−2 s−2 j (k − 1)! j − k k=2 Inserting the explicit form 1 j(s − 1) + 1 (j(s − 1))! cj = = j(s − 1) + 1 j j!(j(s − 2) + 1)! produces j(s−2) (j(s − 1))! j j X 1 k 1 (j(s − 1))! bn+1 = − a + a (−1) j! s−2 s−2 j (k − 1)! (j − k)! k=2 j(s−2) (j(s − 1))! j X k 1 = a −1 + (j − 1)!(−1) . (4.11) j! s−2 (k − 1)!(j − k)! k=2 90 Assume j > 1, then the sum indeed equals one, j(s−2) j(s−2)−1 X 1 X (j − 1)! (j − 1)!(−1)k = − (−1)k (k − 1)!(j − k)! k!(j − 1 − k)! k=2 k=1 j(s−2)−1 X j − 1 = − (j − 1)!(−1)k k k=1 j−1 j(s−2)−1 X j − 1 X j − 1 = − 1j−1−k(−1)k − (−1)k k k k=1 k=j | {z } =0 j−1 X j − 1 j − 1 = − 1j−1−k(−1)k + 1j−1(−1)0 k 0 k=0 = −(1 − 1)j−1 + 1. The value 1 is inserted in eq. (4.11) and yields (j(s − 1))! j bn+1 = a (−1 + 1) = 0. j! s−2 Now assume j = 1. Then, by lemma A.14, Bn,k(...) = (s − 2)!δk1 and the sum in eq. (4.10) vanishes. Since we assumed n = j(s − 2), the only remaining term is bn+1 = bs−1 = −(s − 1)!as−2. The latter statement is not needed for the proof since we only need to consider n ≥ s. But it reproduces lemma 4.3. Finally, let n be no multiple of s − 2. By lemma A.14, all Bn,k(...) vanish and eq. (4.10) becomes (n + 1)! bn+1 = (−n!an). n! By assumption, an is zero and hence bn+1 = 0 in this case, too. 0 Note that setting j = 0 in lemma 4.4 produces aj(s−2) = a0 = c0as−21 = 1 which is the correct result since the diffeomorphism is tangent to identity. Therefore, the formula s stated in this lemma can be used even for a0. The generalization of theorem 3.2 to φ - type interaction is 91 Theorem 4.2. Let s ≥ 3 be an integer and φ a scalar quantum field with mass m and φs-type interac- tion according to eq. (4.1). Assume that all tadpole graphs vanish. Then the quantum field ρ formally defined by λ ρ(x) = φ(x) − φs−1(x). (s − 1)!(p2 − m2) has a free amputed 2-point function ρ(p)ρ(−p) = −i(p2 − m2 + i0) for a fixed external 4- momentum p if it fulfills λφs−2 < 1. (s − 1)!(p2 − m2) Proof. By lemmas 3.3 to 3.5 the corrections to the 2-point function of ρ are cancelled if λ as−2 = = α (s − 1)!(p2 − m2) according to lemma 4.3 and bn = 0 for all n ≥ s. By lemma 4.4 this can be reached by setting j aj·(s−2) = cjα , j ∈ N0 ak = 0, k∈ / N · (s − 2). With this choice of parameters, the diffeomorphism eq. (4.2) is ∞ ∞ X n+1 X j(s−2)+1 φ = anρ = aj(s−2)ρ n=0 j=0 ∞ ∞ X j j(s−2) X s−2j = ρ cjα ρ = ρ Aj(s − 1, 1) αρ j=0 j=0 = ρF αρs−2, (4.12) where F (t) = Cs−1,s(t) is the generating function eq. (1.6) of the Fuss Catalan numbers Aj(s − 1, 1). It fulfills the functional equation eq. (1.7), F (t) − 1 = tF (t)s−1. (4.13) With the substitution h(w(t)) = w(t) + 1 = F (t), eq. (4.13) becomes w(t) = t(w(t) + 1)s−1 = tg(w(t)), s−1 −1 t where we introduced the function g(t) = (t + 1) . The inverse of w(t) is w (t) = g(t) −1 w(t) tg(w(t)) because then w (w(t)) = g(w(t)) = g(w(t)) = t. t t w−1(t) = = . g(t) (t + 1)s−1 92 Since F (t) = h(w(t)), we have w−1 h−1(F (t)) = t and hence t − 1 F −1(t) = w−1 h−1(t) = w−1(t − 1) = . (4.14) ts−1 Define a function 1 t − 1 s−2 z(t) = . (4.15) αt If we use eq. (4.13), we find 1 F (αρs−2) − 1 s−2 z F (αρs−2) = αF (αρs−2) 1 αρs−2F (αρs−2)s−1 s−2 = αF (αρs−2) = ρF (αρs−2) = φ. In the last line eq. (4.12) was used. The equation φ = z(F (αρs−2)) implies 1 1 s−2 ρ = F −1 z−1(φ) . (4.16) α Inverting eq. (4.15) produces t = (1 − αz(t))2−s. The inverse function of z(t) is 1 z−1(x) = . (4.17) 1 − αxs−2 This function only exists for αxs−2 < 1, this is the condition required in the theorem. Note F −1(t) from eq. (4.14) has a pole for t → 0, but z−1(0) = 1 =6 0, so for small enought x the expression F −1(z−1(x)) is not singular. Using eqs. (4.14) and (4.17), equation (4.16) becomes 1 1 1 s−2 ρ = F −1 α 1 − αφs−2 1 1 s−2 s−2 − 1 1 1−αφ = s−1 α 1 1−αφs−2 1 1 s−2 s−2 = 1 − (1 − αφs−2) 1 − αφs−2 . α 93 Note that the condition λρ 1 ≥ − p2 − m2 2 from theorem 3.2 is not the same as λφs−2 < 1 (s − 1)!(p2 − m2) from theorem 4.2 which for s = 3 reduces to λφ < 2. p2 − m2 Actually, all the auxiliary functions used in the proof of theorem 4.2 should be ex- amined for well-definedness and allowed parameter intervals. This is why the result of theorem 4.2 is called “formally”: For the theorem to hold in the sense of ordinary functions, probably additional conditions regarding the allowed parameter intervals of the functions have to be fulfilled. Example 16: Adiabatic diffeomorphism for φ4-theory 4 For φ , s = 4 and hence the function F (t) is C3,1(t) given in eq. (1.14): r q r q 3 1 27αρ2−4 3 1 27αρ2−4 − 2αρ2 + 108α3ρ6 + − 2αρ2 − 108α3ρ6 α < 0 F (αρ2) = √ r q √ r q −1+i 3 3 1 27αρ2−4 1+i 3 3 1 27αρ2−4 2 − 2αρ2 + 108α3ρ6 − 2 − 2αρ2 − 108α3ρ6 α ≥ 0. Here λ α = 6(p2 − m2) is the value a2 computed explicitly in example 15. 4.4 Multiple interaction monomials 4.4.1 Feynman rules The results for φs-theory can easily be generalized to the case of more than one inter- action monomial in the original Lagrangian density. Consider an arbitrary interacting Lagrangian density, as opposed to above it may contain arbitrary many interaction monomials, ∞ 1 µ 1 2 2 X λs s L(x) = − φ(x)∂µ∂ φ(x) − m φ (x) − φ (x). (4.18) 2 2 s! s=3 94 This gives rise to infinitely many families of interaction vertices (s) −iwn = −iλsBn,s(1!a0, 2!a1, 3!a2, 4!a3,...). (4.19) (s) Since for λs the smallest possible vertex has valence s, it must be wn = 0 if n < s. For a given valence n, the interaction monomials give rise to the vertex Feynman rule ∞ X (s) −iwn = −i wn s=3 n X = −i λsBn,s(1!, 2!a1, 3!a2,...). (4.20) s=3 In total, if a diffeomorphism is applied to eq. (4.18), vertices of interaction type −iwn and of pure diffeomorphism type ivn are present at any order. By eqs. (2.15) and (4.20), the general form of a (n ≥ 3)-valent vertex is ivn − iwn = i Bn−2,1(2!a1, 3!a2,...) + Bn−2,2(2!a1, 3!a2,...) (x1 + ... + xn) 2 + i nBn−2,1(2!a1, 3!a2,...) + nBn−2,2(2!a1, 3!a2,...) − Bn,2(1!a0, 2!a1,...) m n X − i λsBn,s(1!, 2!a1, 3!a2,...). (4.21) s=3 Note that the mass term resembles a 2-valent interaction vertex in that it has amplitude Bn,2(1!a0, 2!a2,...). The other two contributions to the mass term arise from the fact p2 x p2 − m2 that we changed j to j = j in the “kinetic” term. 4.4.2 Tree sums Since more than one coupling constant λ is present, the diffeomorphism obtains multiple interaction tree sums like eq. (4.5), each of them is linear in the corresponding coupling constant, n (s) X Sn = −iλs Bn,k(b1, b2,...)Bk,s(1, 2!a1, 3!a2, 4!a3,...). (4.22) k=s It is safe to consider them independently. Any tree containing more than one type of interaction, i.e. a tree proportional to λiλj for some i, j, decomposes. This is because the vertices −iwn never cancel adjacent edges, it is the same mechanism which allows 2 (s) to ignore trees ∝ λ as discussed in section 3.2.4. Clearly, Sn = 0 if n < s, therefore the sum of all trees proportional to any of the coupling constants is n n n X (s) X X Sn = Sn = −i λs Bn,k(b1, b2,...)Bk,s(1, 2!a1, 3!a2, 4!a3,...). (4.23) s=3 s=3 k=s 95 By linearity, theorem 4.1 immediately generalizes to Theorem 4.3. If Sn is the tree sum from eq. (4.23), then Sn = −iλn for any n ≥ 3. Proof. Sn is a sum of terms, each of which is proportional to some λs. By theorem 4.1, only the term where s = n is nonzero and it contributes −iλn. The diffeomorphism ρ has the same S-matrix as the underlying theory φ. 96 5 Properties of the adiabatic diffeomorphism 5.1 ρ as a free field 5.1.1 ρ in momentum space By construction, ρ has, for a given offshell variable xp =6 0, a free 2-point function. A priori, its other properties are unknown. Here we argue that it indeed is “almost free”. The adiabatic diffeomorphism from theorem 4.2 reads, after multiplication with xp, λ p2 − m2ρ(x) = p2 − m2φ(x) − φs−1(x). (5.1) (s − 1)! By Fourier transformation, a product becomes a convolution Z d4x Z φ2(x)eikx = d4q φ(q)φ(k − q) = φ∗2(k), (2π)4 hence the product φs−1(x) becomes a (s − 1)-fold convolution and eq. (5.1) reads in momentum space λ p2 − m2ρ(k) = p2 − m2φ(k) − φ∗(s−1)(k). (5.2) (s − 1)! Example 17: φ3-theory For s = 3, λ p2 − m2ρ(k) = p2 − m2φ(k) − φ∗2(k) 2 λ Z = p2 − m2φ(k) − d4q φ(q)φ(k − q). 2 This holds for arbitrary k just as the original form was true for any spacetime point x. Especially, one can consider k = p. From this point of view, it seems very tempting to change p → k in eq. (5.4) to obtain λ ρ(k) = φ(k) − φ∗(s−1)(k) for any k. (5.3) (s − 1)!(k2 − m2) 97 This seemingly small change has drastic consequences: Now it is no longer possible to undo the Fourier transform by replacing ∗ → ·, instead the second summand has a k-dependent prefactor which also needs to be taken into the transformation. The field ρ(x) obtained in this way is no longer a local function of φ(x), but depends on φ(y) for y =6 x. Therefore eq. (5.3) is not a diffeomorphism in the sense considered in this work. If the property “free 2-point function at fixed momentum p” can be extended to “all momenta k” this way, the field ρ consequently has a free propagator altogether. The Jost-Schroer theorem 1.1 ensures that a free 2-point function is sufficient for a field to be free altogether, but care must be taken because the theorem is about Wightman- distributions whereas ρ has a free time-ordered 2-point function. Whether the Jost- Schroer theorem also allows for a statement in this case needs to be examined in future work. Example 18: Explicit transformation formula for φ3 If λ Z ρ(k) = φ(k) − d4q φ(q)φ(k − q) 2(k2 − m2) then λ Z Z e−ikxφ(q)φ(k − q) ρ(x) = φ(x) − d4k d4q 2 k2 − m2 λ Z Z φ(k − q)e−ikx = φ(x) − d4q φ(q) d4k . 2 k2 − m2 5.1.2 Classical fields With the Lagrangian density eq. (4.1), 1 µ 1 2 2 λ s L(x) = ∂µφ(x)∂ φ(x) − m φ (x) − φ (x), 2 2 s! the classical equations of motion (i.e. the Euler-Lagrange-equations) are µ 2 λ s−1 ∂µ∂ + m φ(x) + φ (x) = 0. (s − 1)! In the frequency domain this reads λ k2 − m2φ(k) − φ∗(s−1)(k) = 0. (5.4) (s − 1)! Assume φ(x) is a classical field, i.e. it fulfills eq. (5.4) for any k, then especially for the fixed value p we have λ p2 − m2φ(p) − φ∗(s−1)(p) = 0 for the fixed momentum p. (5.5) (s − 1)! 98 Consequently the adiabatic diffeomorphism ρ(k) defined by eq. (5.2) fulfills p2 − m2ρ(p) = 0 for the fixed momentum p. This is the Euler-Lagrange-equation of a free field, i.e. “ρ is a classical free field for the fixed momentum p”. If one “continues” the definition of ρ to arbitrary momenta as proposed in eq. (5.3) then if φ was a classical φ3-field, ρ would become a classical free field altogether, that is k2 − m2ρ(k) = 0 for any momentum k. 5.1.3 Cancellation of higher correlation functions But besides the fact that ρ appears to mimic a free field on the classical level, also as a quantum field more than just the 2-point function cancels. This is due to the fact that ρ(x) by lemma 4.4 is chosen such that bn = 0∀n. Since these bn are just the Feynman amplitudes of tree sums, we can conclude immediately: The treelevel- amplitudes where one external momentum is p and all other external momenta are onshell vanish. Probably, this also extends to loop momenta by the same argument as section 2.4: Any loop amplitude can be reconstructed from cuts which eventually depend on trees, as long as precisely one external edge of the overall amplitude has momentum p, there will always be one tree in the products which has this edge and consequently vanishes, hence all products vanish and so do all loop amplitudes. By this, the statement “ρ is free for momentum p” becomes sensible at quantum level: Any connected n-point function vanishes given at least one of the momenta is p. Recall that if all of the momenta are onshell (and none is equal to p), the n-point functions are the ones of usual φs-theory by theorem 4.3. 5.2 An identity for φs-theory Equation (5.2) is an explicit function for ρ(k). Since we know that by definition ρ has a free 2-point function, one immediately has −i p2 − m2 = ρ(p)ρ(−p) λ λ = φ(p) − φ∗(s−1)(p) φ(−p) − φ∗(s−1)(−p) (s − 1)!xp (s − 1)!xp λ = φ(p)φ(−p) − φ(p)φ∗(s−1)(−p) + φ(−p)φ∗(s−1)(p) (s − 1)!xp 2 λ ∗(s−1) ∗(s−1) + 2 φ (p)φ (−p). (5.6) ((s − 1)!xp) 99 Example 19: Identity for φ3-theory For φ3-theory, i.e. s = 3, eq. (5.6) reduces to 2 λ ∗2 ∗2 λ ∗2 ∗2 −ixp = φ(p)φ(−p) − φ(p)φ (−p) + φ(−p)φ (p) + 2 φ (p)φ (−p) 2xp 4xp = φ(p)φ(−p) λ Z Z − φ(p) d4q φ(q)φ(−p − q) + φ(−p) d4q φ(q)φ(p − q) 2xp 2 Z Z λ 4 4 + 2 d k φ(k)φ(p − k) d q φ(q)φ(−p − q) 4xp λ Z D E = φ(p)φ(−p) − d4q φ(p)φ(q)φ(−p − q) + φ(−p)φ(q)φ(p − q) 2xp 2 ZZ λ 4 4 D E + 2 d kd q φ(k)φ(p − k)φ(q)φ(−p − q) . (5.7) 4xp This identity involves correlation functions up to order (s − 1)2. Note that the convolution product automatically implements momentum conservation, hence these correlation functions are taken at sensible kinematic points. Example 20: First order in φ3-theory In φ3-theory, Z 1 2 4 i i 4 φ(p)φ(−p) = −ixp + i (−iλ) d q + O λ , 2 xq xp+q φ(p)φ(q)φ(−p − q) = −iλ + O λ3, i i i φ(k)φ(p − k)φ(q)φ(−p − q) = (−iλ)2 + + + O λ4, xp xp+q−k xq+k hence eq. (5.7) reads Z Z 1 2 4 1 λ 4 4 −ixp = −ixp + i λ d q − d q(−iλ − iλ) + O λ . 2 xqxp+q 2xp At order λ2 the equation implies Z 1 d4q + 1 = 0. xqxp+q This requirement can seemingly only be fulfilled if p is chosen as the renormalization point of φ, that is, if all Feynman graphs vanish at this point. So far it is dubious if eq. (5.6) is a valid statement and how to make sense of it. 100 5.3 Conclusion and outlook We have established two results for scalar quantum fields with interaction: 1. For arbitrary interaction polynomials of the underlying field, diffeomorphisms do not alter onshell tree sums (theorem 4.3). In kinematic renormalization this implies they do not alter the S-matrix and are therefore not observable in exper- iments. In this sense, all quantum fields which are related by diffeomorphisms form an equivalence class. This opens the opportunity to pick out of this equiv- alence class any representative with desirable properties for the problem under consideration. 2. If only one interaction monomial (of arbitrary order) is present, then there is a diffeomorphism which for specific fixed offshell external momenta posesses a free time-ordered 2-point function and it is explicitly given by theorem 4.2. Still, a number of questions remains to be adressed in future work: 1. What are the conditions for the function ρ(φ) to exist in a mathematically rigorous way? Especially, if there are constraints on the fields as in theorem 3.2, are they fulfilled by quantum fields? 2. φs-theory has s-valent vertices in Feynman graphs. On the other hand, the adiabatic diffeomorphism by lemma 4.4 is given by the Fuss-Catalan numbers An(s − 1, 1). These numbers in turn count the planar trees built of s-valent ver- tices. Is there a deeper combinatoric meaning in the adiabatic diffeomorphism which is not yet understood? 3. What is the status of ρ? Does passing from φ to ρ effectively just implement kinematic renormalization at the renormalization point p? What does this even mean for the non-renormalizable theory ρ? 4. Is it possible to map from an interacting field φ to a completely free one by the nonlocal transformation eq. (5.3)? 5. Are the identities noted in section 5.2 meaningful in any way? Are they even correct? 6. Is it actually possible to use the adiabatic diffeomorphism to cure at least in part the ill-definedness of perturbative quantum field theory, and if so, how? 7. If any set of (possibly infinitely many) vertex Feynman rules is given, is there a way to determine whether this is just a diffeomorphism of some simpler theory? 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(A.1) holds, n k−1 X X Xn := xj+k = x1+2 + x1+3 + ... + x1+n + ... + x(n−1)+n (A.2) k=2 j=1 n X n(n − 3) = (n − 2) x + m2. k 2 k=1 n(n−1) Proof. Note that there are 2 summands in eq. (A.2). For any n ≥ 2, n k−1 n k−1 2 X X 2 X X 2 2 n(n − 1) 2 Xn = p + p − m = p + p + 2p p − m j k j k j k 2 k=2 j=1 k=2 j=1 Rearranging summation indices and using momentum conservation yields for the indi- 107 vidual parts n k−1 n k−1 n n n k−1 X X X X X X X X p2 p2 p2 p2 j + k = j 1 + k 1 k=2 j=1 k=2 j=1 j=1 k=j+1 k=2 j=1 n n n X X X n − k p2 k − p2 n − p2 = ( ) k + ( 1) k = ( 1) k k=1 k=2 k=1 n k−1 n k−1 n n X X X X X X p p p p p p 2 k j = k j + j k k=2 j=1 k=2 j=1 j=1 k=j+1 n n X X X p p p −p , = k j = k k k=1 j6=k k=1 so n X 2 2 n(n − 1) 2 Xn = (n − 2) p − (n − 2)nm + (n − 2)n − m k 2 k=1 Lemma A.2. If in in the sum from lemma A.1 still momentum conservation eq. (A.1) between all n p momenta but all variables involving n are left out, then n−1 k−1 0 X X Xn := xj+k = x1+2 + x1+3 + ... + x1+(n−1) + ... + x(n−2)+(n−1) (A.3) k=2 j=1 n−1 X (n − 2)(n − 3) 2 = (n − 3) xi + xn + m . 2 i=1 Proof. n−1 X X0 X − p2 p2 p p − m2 n = n j + n + 2 j n j=1 n−1 n−1 X X X − x − n − x − p p − n − m2 = n j ( 1) n 2 n j ( 1) j=1 j=1 n X X − x − n − x − p −p − n − m2 = n j ( 2) n 2 n n ( 1) j=1 n n X n(n − 3) 2 X 2 = (n − 2) x + m − xj − (n − 2)xn + 2xn − (n − 3)m k 2 k=1 j=1 n X n(n − 3) 2 2 = (n − 3) x − (n − 4)xn + m − (n − 3)m . k 2 k=1 108 Note that this is not the same as Xn−1 because the latter involves momentum con- n − p servation only between the first ( 1) momenta, which effectively implies n = 0. Example 21: n = 4 For n = 4, the lemmas yield 2 X4 = 2(x1 + x2 + x3 + x4) + 2m 0 2 X4 = x1 + x2 + x3 + x4 + m . 0 By chance, 2X4 = X4. This is true only for n = 4 because in this case, due to X p momentum conservation, the offshell variables in 4 coincide pairwise, e.g. 1 + p −p − p x x 2 = 3 4 implies 1+2 = 3+4. A.2 Compatibility of symmetry factors Let Γ be any Feynman graph. Definition 10. A Cutkosky cut C of Γ is a set of internal edges EΓ,C ⊂ EΓ of Γ to produce k subgraphs Γ1,..., Γk ⊂ Γ such that cutting any smaller set x ⊂ EΓ,C produces stricty less subgraphs. Now fix a graph Γ and a Cutkosky cut C. Then each Γi has some external edges xi which were external to Γ and some edges ei,j which are external to Γi but used to connect Γi to Γj in Γ. So connecting all edges ei,j to their partners in ej,i reproduces Γ. But connecting them in another way will generally produce a graph different from Γ. There are Y ei,j! i 109 Lemma A.3. In the sum X 1 F ({Γ1,..., Γk}), (A.5) sym(Γ1) ··· sym(Γk) {Γ1,...,Γk} compatible with (Γ,C) 1 the graph Γ appears and is weighted with sym(Γ) mul(Γ,C) where mul(Γ,C) is the num- ber of Cutkosky cuts C which are different if the edges of Γ are labelled but produce the same subgraphs if the labeling is forgotten. Proof. This is [KY17, Proposition 4.1], an abstract proof can be found there. The principle can be understood if one considers a cut into just two connected components. The symmetry factor of Γ is determined by three different contributions. 1. Permutations in Γ which are not touched by the cut C. These permutations thus are also possible in one of the Γi and are encoded in sym(Γi). 2. Permutations of edges in Γ which are cut. Let there be e.g. n equivalent edges, then sym(Γ) obtains a factor n! from their permutation. But if Γ is cut, each of the two Γi has these n edges in ei,j and thus each of the sym(Γi) contains a factor 1 n!, giving (n!)2 in the sum eq. (A.5). But there are precisely n! possiblilites to connect the edges of e1,2 to those of e2,1 such that Γ is reproduced, hence eq. (A.5) 1 1 contains n! summands with a prefactor (n!)2 each, reproducing n! . 3. Permutations of parts (i.e. vertices or bigger subgraphs) of Γ which were possible in Γ but are no longer possible if Γ is cut. For the case of just two connected components, this means there are two exchangeable subgraphs in Γ, one of which ends up in Γ1 and the other one in Γ2. In Γ, the interchangeabiligy produced a symmetry factor sym(Γ) = 2, but the symmetry factor of the Γi is not altered. But in this case, there is necessarily an equivalent Cutkosky cut C0 where the subgraphs are assigned to the respective other Γi. This means mul(Γ,C) = 2. Since the total symmetry factor is the product of the symmetry factors from permuta- tions of individual parts, it will be reproduced correctly. 110 Example 22: 2-loop graphs Consider 2-loop graphs of the 2-point function. If one chooses Γ1 and Γ2 as shown below, there are six possible ways of connecting their external edges 1, 2, 3 and 4, 5, 6, respectively. 1 ΓA = 2 Γ1 = 3 ΓB = 4 5 Γ2 = 6 ΓC = It is sym(Γ1) = 2 = sym(Γ2) since e.g. edges 1 and 2 can be interchanged. So the 1 1 1 sum from eq. (A.5) produces two times the topology ΓC , giving 2 2 2 = 2 , which is the correct symmetry factor. But the other topology is produced four times, 1 1 1 2 2 4 = 1, whereas sym(ΓA) = 2 . Indeed, this is twice the expected result since two equivalent Cutkosky cuts are possible as indicated in ΓA and ΓB. Note that these cuts differ only if the edges of the graph are labelled i.e. it is considered a planar graph. This is normally not the case in QFT. A.3 Combinatoric Lemmas Lemma A.4. k k−l k k−l X X 1 X X (1 + k − l − m)c cmc = (k + 3)c cmc . l k−l−m 3 l k−l−m l=0 m=0 l=0 m=0 Proof. Let the lhs be V . Replacing k − l − m → u by setting m = k − l − u and finally u → m produces k k−l X X V = (1 + m)clcmck−l−m. l=0 m=0 111 Pk Pk−u Pk Pk−m Now l → u and u=0 m=0 → m=0 u=0 and then renaming m → l and u → m this is k k−l X X V = (1 + l)cmclck−m−l. l=0 m=0 This gives three different forms of V . On the other hand k k−l X X V = (1 + k − l − m)clcmck−l−m l=0 m=0 k k−l k k−l X X X X = (3 + k − (1 + l) − (1 + m))clcmck−l−m = (k + 3)clcmck−l−m − 2V l=0 m=0 l=0 m=0 and hence k k−l X X 3V = (k + 3)clcmck−l−m. l=0 m=0 Lemma A.5. P∞ n P∞ n If D(t) = n=0 dnt and C(t) = n=1 cnt and one sets xn cn = , n ≥ 1 n! n X (s + l)! dn = B (x1, x2, x3,...), n ≥ 0 n!s! n,l l=0 then 1 s+1 D(t) = . 1 − C(t) Proof. By eq. (1.17) ∞ n ∞ ∞ X X tn(s + l)! X (s + l)! X tn D(t) = B (x1, x2,...) = B (x1, x2,...) s!n! n,l s! n! n,l n=0 l=0 l=0 n=l l ∞ ∞ X (s + l)! X tj = xj . s!l! j! l=0 j=1 The parenthesis is ∞ ∞ X xj j X j t = cjt = C(t), j! j=1 j=1 112 so ∞ X (s + i)! D(t) = Ci(t) s!i! i=0 ∞ X s + i = Ci(t). s i=0 Binomial coefficients (with integer upper entry) fulfill n −n + i − 1 = (−1)i , i i this allows to use the binomial series, ∞ X −(s + i) + i − 1 D(t) = (−1)iCi(t) i i=0 ∞ X −(s + 1) = (−C)i(t) i i=0 1 s+1 = . 1 − C(t) Several identities follow from lemma A.5. To avoid unreadable notation with the sum as series coefficient dn, retreat first to the generic case. Assume C(t) and D(t) are functions fulfilling the following conditions: ∞ X n C(t) = cnt , c0 = −1 n=1 ∞ X n D(t) = dnt (A.6) n=0 1 s+1 and D(t) = . 1 − C(t) Lemma A.6. Under the conditions eq. (A.6) either s = 2 or n k−3 k−3−l X X X kclcmck−3−l−mdn−k = 0. (A.7) k=3 l=0 m=0 Proof. This proof follows a technique used in [Cvi13]. The derivative of D(t) fulfills D0(t)(1 − C(t)) = (s + 1)D(t)C0(t). (A.8) 113 Multiplying by (1 − C(t))2 yields D0(t)(1 − C(t))3 = (s + 1)D(t)C0(t)(1 − C(t))2. (A.9) The series expansion of the left hand side is ∞ s s−l 3 X X X s (1 − C(t)) = − clcjcs−l−mt . s=0 l=0 m=0 ∞ u k k−l 0 3 X X X X u −D (t)(1 − C(t)) = (u + 1 − k)clcmck−l−mdu+1−kt . u=0 k=0 l=0 m=0 n−4 k k−l n−4 0 3 X X X − t D (t)(1 − C(t)) = (n − 3 − k)clcmck−l−mdn−3−k k=0 l=0 m=0 Here, k = n − 3 can be included since (n − 3 − k) is zero in this case, hence n−3 k k−l n−4 0 3 X X X − t D (t)(1 − C(t)) = (n − 3 − k)clcmck−l−mdn−3−k. k=0 l=0 m=0 The right hand side of eq. (A.9) is ∞ u v 2 0 X X X u (1 − C(t)) C (t) = (u − v + 1)cicv−icu−v+1t u=0 v=0 i=0 ∞ u k k−l 2 0 X X X X u D(t)(1 − C(t)) C (t) = (k − m − l + 1)clcmck−m−l+1du−kt . u=0 k=0 l=0 m=0 The order tn−4 of this term hence is n−3 k−1 k−1−l n−4 2 0 X X X t D(t)(1 − C(t)) C (t) = (k − l − m)clcmck−l−mdn−3−k. k=1 l=0 m=0 It is possible to include m = k −l and also l = k into the summation since in both cases the prefactor (k − l − m) vanishes. Even k = 0 can be included, n−3 k k−l n−4 2 0 X X X t D(t)(1 − C(t)) C (t) = (k − l − m)clcmck−l−mdn−3−k. k=0 l=0 m=0 eq. (A.9) now implies 0 = −tn−4D0(t)(1 − C(t))3 + (s + 1)tn−4D(t)(1 − C(t))2C0(t) n−3 k k−l X X X = (s + 1)(1 + k − l − m)clcmck−l−mdn−3−k k=0 l=0 m=0 n−3 k k−l X X X − (k + 3)clcmck−l−mdn−3−k. k=0 l=0 m=0 114 By lemma A.4 both sums are the same up to a prefactor, n−3 k k−l 1 X X X 0 = (s + 1) − 1 (k + 3)c cmc d . 3 l k−l−m n−3−k k=0 l=0 m=0 Then if s =6 2 n−3 k k−l X X X (k + 3)clcmck−l−mdn−3−k = 0, k=0 l=0 m=0 this is eq. (A.6) with shifted summation indices. Lemma A.7. n−r+1 X xw Bn−w,r−1(x1, x2,,...) Bn,r(x1,...) (w − 1) = (n − r) w! (n − w)! n! w=2 Proof. By [KY17], Lemma 2.4 (which is a modified form of [Cvi13]) it is n X xs Bn−s,r−1(x1,...) Bn,r(x1,...) = r s! (n − s)! n! s=0 n X xs Bn−s,r−1(x1,...) Bn,r(x1,...) s = . s! (n − s)! (n − 1)! s=0 However, it seems in the first equation the sum better start at s = 1 to be valid. Namely, see eq. (1.15), B0,k = 0 ∀k > 0, this is implied in [Cvi13] equation 1.4 but lost in [KY17] when they substitute Bn,1(...) = xn. The problem does not occur in the second equation since s = 0 makes the summand vanish anyway. The correct form thus is n X xs Bn−s,r−1(x1,...) Bn,r(x1,...) = r . s! (n − s)! n! s=1 Combining both above sums directly yields n X xs Bn−s,r−1(x1,...) Bn,r(x1,...) r (s − 1) = 1 − . s! (n − s)! (n − 1)! n s=1 Now By,r−1(...) = 0 if y < r − 1. Thus, all summands vanish where n − s < r − 1 or s > n − r + 1. What remains is the first summand s = 1 but this one equals zero due to the prefactor (s − 1). In lemma A.6, the coefficients clcmck−3−l−m in the summands turned out to be the explicit form of the third power of (1 − C(t)). Using results on coefficients of powers of generating functions the above lemma can indeed be generalized to arbitrary powers. 115 Lemma A.8. If eq. (A.6) holds with s = n − 1 then either n = r or n X 1 B (1!c0, 2!c1, 3!c2,...)d = 0. (k − 1)! k,r n−k k=r Proof. Use D0(t)(1 − C(t)) = (s + 1)D(t)C0(t) and multiply by (1 − C(t))r−1: D0(t)(1 − C(t))r = (s + 1)D(t)C0(t)(1 − C(t))r−1 00 = D0(t)(C(t) − 1)r + (s + 1)D(t)C0(t)(C(t) − 1)r−1. (A.10) Now define ∞ X n f(t) = t(C(t) − 1) = cn−1t n=1 ∞ r X n g(t) = t = δnrt n=0 then, by Fa`adi Bruno‘s formula eq. (1.19) ∞ X 1 n g(f(x)) = r!Bn,r(1!c0, 2!c1, 3!c2,...)x , n! n=1 so [tn−r](C(t) − 1)r) = [tn] (t(C(t) − 1))r 1 = r!Bn,r(1!c0, 2!c1, 3!c2,...) n! Since ∞ 0 X n C (t) = (n + 1)cn+1t n=0 ∞ 0 X n D (t) = (n + 1)dn+1t n=0 it is [tn−1−r] D0(t)(C(t) − 1)r n−1−r X = [tv](C(t) − 1)r[tn−1−r−v]D0(t) v=0 n−1 X = [tk−r](C(t) − 1)r[tn−1−k]D0(t) k=r n−1 X 1 = r!(n − k)B (1!c0, 2!c1, 3!c2,...)d . k! k,r n−k k=r 116 The summand k = n can be included since (n − k) vanishes then. n n−1−r 0 r X 1 [t ] D (t)(1 − C(t)) = r!(n − k)B (1!c0, 2!c1, 3!c2,...)d . (A.11) k! k,r n−k k=r On the other hand ∞ r−1 r−1 X 1 n t (C(t) − 1) = 1 + (r − 1)!Bn,r−1(1!c0, 2!c1, 3!c2,...)t n! n=1 and thus [tn−r+1] (C(t) − 1)r−1 = [tn] tr−1(C(t) − 1)r−1 1 = (r − 1)!Bn,r−1(1!c0, 2!c1, 3!c2,...). n! k−r−1 X [tk−r−1] C0(t)(1 − C(t))r−1 = [tk−r−1−v](1 − C(t))r−1[tv]C0(t) v=0 k−r+1 X = [tk−w−r+1] (1 − C(t))r−1 [tw−2]C0(t) w=2 k−r+1 X 1 = (r − 1)!B (1!c0, 2!c1,...)(w − 1)cw−1. (k − w)! k−w,r−1 w=2 Now introduce k!ck−1 = xk, then k−r+1 X xw Bk−w,r−1(x1, x2, x3,...) [tk−r−1] C0(t)(1 − C(t))r−1 = (r − 1)! (w − 1) . w! (k − w)! w=2 This is lemma A.7, Bk,r(x1,...) [tk−r−1] C0(t)(1 − C(t))r−1 = (r − 1)!(k − r) . k! Finally, [tn−1−r] D(t)C0(t)(1 − C(t))r−1 n−1−r X = [tv] C0(t)(1 − C(t))r−1 [tn−1−r−v]D(t) v=0 n X = [tk−r−1] C0(t)(1 − C(t))r−1 [tn−k]D(t) k=1+r n X Bk,r(x1,...) = (r − 1)!(k − r) d k! n−k k=r+1 117 The summand k = r vanishes and can be included without changing the sum, n X Bk,r(x1,...) [tn−1−r] D(t)C0(t)(1 − C(t))r−1 = (r − 1)!(k − r) d (A.12) k! n−k k=r Now eq. (A.10) implies 0 = [tn−1−r] D0(t)(1 − C(t))r + (s + 1)[tn−1−r] D(t)C0(t)(1 − C(t))r−1 Inserting eqs. (A.11) and (A.12) turns this into n X 1 0 = (r!(n − k) + (s + 1)(r − 1)!(k − r)) B (1!c0, 2!c1, 3!c2,...)d . k! k,r n−k k=r Divide by (r − 1)! and set s = n − 1 as requested in the conditions, n X 1 0 = (r(n − k) + n(k − r)) B (1!c0, 2!c1, 3!c2,...)d k! k,r n−k k=r n X 1 = (n − r) B (1!c0, 2!c1, 3!c2,...)d . (k − 1)! k,r n−k k=r Lemma A.9. Under the conditions eq. (A.6), n i X X 2(s + 1)(i − j)j + (n − i)i dn−ici−jcj = 0 i=0 j=0 n i X X 2(n − i) + (s + 1)i dn−ici−jcj = 0 i=0 j=0 n i X X (s + 1)(i − j)j(i − 2) + (n − i)(j(j − 1) + (i − j)(i − j − 1)) dn−1ci−jcj = 0 i=0 j=0 n i X X (s + 1)i(i − 1) + (n − i)i − (s + 1)(j(j − 1) + (i − j)(i − j − 1)) dn−ici−jcj = 0 i=0 j=0 Proof. This is [KY17, lemma 2.6]. The proof works along the same lines as the one of lemma A.6. Lemma A.10. If x0 = −1 and n =6 3 then n k−3 k−3−l n−k X X X k xlxmxk−3−l−m X (n − 1 + j)!B (x1, x2,...) = 0. (n − k)! l!m!(k − 3 − r − m)! n−k,j k=3 l=0 m=0 j=0 118 Proof. By lemma A.5 the choice n X (s + l)! dn = B (x1, x2,...), n ≥ 0 n!s! n,l l=0 n−k 1 X ⇒ d = (s + j)!B (x1, x2,...), n ≥ k n−k (n − k)!s! n−k,j j=0 xn cn = , n ≥ 1, c0 = x0 = −1 n! fulfills eq. (A.6). Hence, lemma A.6 can be applied, n k−3 k−3−l n−k 1 X X X xlxmxk−3−l−m k X (s + j)!B (x1, x2,...) = 0. s! l!m!(k − 3 − l − m)! (n − k)! n−k,j k=3 l=0 m=0 j=0 Setting s → n − 1 gives the statement, the requirement s =6 2 from lemma A.6 becomes n =6 3. Lemma A.11. With n X γn = (s + l)!Bn−i,l(x1, x2,...) l=0 it is n i X X 2(s + 1)(i − j)j + (n − i)i xi−jxj γn = 0 (n − i)! (i − j)!j! i=0 j=0 n i X X 2(n − i) + (s + 1)i xi−jxj γn = 0 (n − i)! (i − j)!j! i=0 j=0 n i X X (s + 1)(i − j)j(i − 2) + (n − i)(j(j − 1) + (i − j)(i − j − 1)) xi−jxj γn = 0 (n − i)! (i − j)!j! i=0 j=0 n i X X (s + 1)i(i − 1) + (n − i)i − (s + 1)(j(j − 1) + (i − j)(i − j − 1)) xi−jxj γn = 0. (n − i)! (i − j)!j! i=0 j=0 Proof. This is [KY17, lemma 2.8] and a direct consequence of lemma A.9 using lemma A.5. Lemma A.12. If x0 = −1 and n =6 s then n n−k X 1 X (n − 1 + j)! B (1x0, 2x1, 3x2,...) B (x1, x2,...) = 0 (k − 1)! k,s (n − k)! n−k,j k=s j=0 119 Proof. Similar to the proof of lemma A.10, define n−k 1 X d = (n − 1 + j)!B (x1, x2,...), n ≥ k n−k (n − k)!(n − 1)! n−k,j j=0 xn cn = , n ≥ 1, c0 = x0 = −1 n! to fulfill eq. (A.6) with s = n − 1 by lemma A.5 and obtain for the above sum n X 1 B (1 · 0!c0, 2 · 1!c1, 3 · 2!c2,...)(n − 1)!d (k − 1)! k,s n−k k=s n X 1 = (n − 1)! B (1!c0, 2!c1, 3!c2,...)d . (k − 1)! k,s n−k k=s This vanishes by lemma A.8 if n =6 r. Lemma A.13. (2m)! With the Catalan numbers Cm = m!(m+1)! from eq. (1.5), (2n)!(n − 1)! B (1!C1, 2!C2, 3!C3,...) = n,k (k − 1)!(n + k)!(n − k)! Proof. The generating function of the Bell polynomials eq. (1.17) in this case is ∞ n n ∞ j X X t k X t Bn,k(1!C1, 2!C2, 3!C3,...) u = expu j!Cj n! j! n=0 k=0 j=1 ∞ X j = expu Cjt j=1 ∞ X j 0 = expu Cjt − C0t j=0 = exp(u(C(t) − 1)) ∞ X 1 = (u(C(t) − 1))k k! k=0 where ∞ X j C(t) = Cjt j=0 is the generating function of the Catalan numbers eq. (1.6). Changing summation order on the lhs, this reduces to an equation for the summands, n! n k B (1!C1, 2!C2, 3!C3,...) = [t ](C(t) − 1) . n,k k! 120 On the rhs, apply the functional equation eq. (1.9) to obtain n! n 2 k B (1!C1, 2!C2, 3!C3,...) = [t ] tC (t) n,k k! n! = [tn−k]C2k(t). k! Now apply lemma 1.1 with q = 2, b = 2k, h i 2k 2m + 2k tn−k C2k(t) = , 2m + 2k m m=n−k hence n! k 2n B (1!C1, 2!C2, 3!C3,...) = . n,k k! n n − k Lemma A.14. For m ∈ N, with Fuss-Catalan numbers cj = Aj(m+1, 1) from definition 2 and if n > m is a multiple of m then m−1 times m−1 times n z }| { z }| { (n − 1)! m (m + 1) Bn,k 0, 0,..., 0, m!c1, 0,..., 0 , (2m)!c2, 0,... = m n . (k − 1)! m − k If n = m then Bn,k(0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) = m!δk1, for all other n Bn,k(0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) = 0. Proof. Note lemma A.13 is the special case of this lemma for m = 1 and the proof therefore works similarly. ∞ ∞ n X X t k B (0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) u n,k n! k=0 n=k j ∞ X t X mj = expu j!c j = expu cjt m j! j∈N·m j=1 ∞ X m j = expu cj(t ) − 1 j=0 m = exp(u(Cm+1,1(t ) − 1)) ∞ X 1 k m k = u (Cm+1,1(t ) − 1) k! k=0 121 Here, Cm+1,1 is the generating function of the Fuss-Catalan numbers from eq. (1.6). The equation holds for each summand, ∞ X tn B (0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) n,k n! n=k 1 m k = (Cm+1,1(t ) − 1) k! By eq. (1.7) m+1 Cm+1,1(t) − 1 = t · Cm+1,1(t) and therefore Bn,k(0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) n! n m k = [t ](Cm+1,1(t ) − 1) k! n! k = [tn] tmCm+1 (tm) k! m+1,1 n!h i k(m+1) = tn−mk C (tm). k! m+1,1 The series coefficient follows from lemma 1.1 with q = m + 1, b = k(m + 1), Bn,k(0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) ∞ n!h i X k(m + 1) s(m + 1) + k(m + 1) = tn−mk (tm)s . (A.13) k! s(m + 1) + k(m + 1) s s=0 The result is zero unless n − mk = ms for some s ∈ N. Since also k ∈ N, this implies n n has to be a multiple of m. Assume also n > 1m. Under this condition s = m − k and Bn,k(0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) n n! k(m + 1) m − k (m + 1) + k(m + 1) = n n k! m − k (m + 1) + k(m + 1) m − k If on the other hand n = 1m then s = 1 − k, but in the sum eq. (A.13) still s ≥ 0, therefore only k = 1 gives a non-vanishing result, m 0 m + 1 m + 1 Bm,1(0,..., 0, m!c1, 0,..., 0, (2m)!c2, 0,...) = m!(t ) = m!. m + 1 0 This can also be seen immediately from eq. (1.15). Lemma A.15. n X (n + k)! bn+1 = B (−1!a1, −2!a2,...) n! n,k k=0 n X (m − 1 + k)! ⇔ Bm,m−n(b1, b2,...) = B (−1!a1, −2!a2,...) m > n =6 0. (m − 1 − n)!n! n,k k=0 Proof. This is [KY17], Lemma 3.2. The proof is given there. 122 B Umgangssprachliche Erl¨auterungder Arbeit Diese Masterarbeit ist stellenweise etwas technisch und fur¨ Laien schwer verst¨andlich. Deshalb folgt hier eine umgangssprachliche Erl¨auterung der Problemstellung und der zentralen Ergebnisse fur¨ Nichtphysiker anhand einer Analogie. Aus der Alltagserfahrung weiß man, dass es keine Wechselwirkung von Licht mit sich selbst gibt: Die Strahlen mehrerer Laserpointer prallen nicht aneinander ab, sondern durchdringen sich und sind danach unver¨andert. Wenn man mehrere gleichartige Lam- pen nacheinander anschaltet, wird der Raum im gleichen Maße heller. Mathematisch wurde¨ man sagen, Licht verh¨alt sich wie ein freies Feld. Dabei lassen wir außen vor, dass es naturlich¨ Wechselwirkung zwischen Licht und Materie gibt, schließlich durchdringt der Laserpointer keine W¨ande, und konzentrieren uns nur auf den Aspekt von Licht mit Licht. Angenommen, wir wollen durch ein Experiment uberpr¨ ufen,¨ ob Licht ein freies Feld ist. Ein m¨oglicher Aufbau w¨are ein Lichtsensor, auf den wir nach und nach mit meh- reren gleichen Lampen leuchten. Wir werden feststellen, dass die Lichtintensit¨at mit jeder neuen Lampe um einen festen Wert zunimmt. Daraus schließen wir, dass Licht tats¨achlich ein freies Feld ist. Es k¨onnte aber sein, dass unser Lichtsensor minderwertig und seine Skala nicht gleichm¨aßig unterteilt ist. Dann wurde¨ bei zwei Lampen zwar nach wie vor doppelt so viel Licht einfallen wie bei einer, aber der Sensor wurde¨ nicht den doppelten Mess- wert anzeigen. Wenn wir so einen Sensor verwenden, dann kommen wir zu dem falschen Schluss, dass das Licht der beiden Lampen sich offensichtlich gegenseitig beeinflusst und dass Licht folglich kein freies Feld ist. Aus diesem Gedankenexperiment ist ersichtlich, dass die Verwendung einer krum- ” men“ Skala im Allgemeinen dazu fuhrt,¨ dass ein damit gemessenes Feld so erscheint, als habe es eine Wechselwirkung mit sich selbst. Man spricht vereinfacht von einem wechselwirkenden Feld. Andersherum ist auch plausibel, dass man fur¨ ein tats¨achlich wechselwirkendes Feld eine Skala finden kann, auf der es wie ein freies Feld erscheint. Diese Masterarbeit behandelt das analoge Ph¨anomen auf Basis der Quantenfeldtheo- rie. Messungen in der Quantentheorie sind stets konzeptionell schwierig, wir betrachten daher nicht den Fall, dass ein Feld gemessen wird, sondern wir konzentrieren uns dar- auf, wie sich die Rechnungen selbst ver¨andern, wenn man ein Feld durch eine krumme“ ” Funktion dieses Feldes ersetzt. Unser Feld heißt φ (Phi) und ist ein Skalarfeld, das ist ¨ahnlich wie Licht, aber mathematisch einfacher zu behandeln, weil es nur eine m¨ogliche Polarisationsrichtung hat (Licht hat zwei). In der Natur gibt es kein Skalarfeld in der genauen Form, wie wir es hier benutzen. Lediglich das Higgs-Boson ist ein Skalarfeld, es hat aber noch zus¨atzlich Wechselwirkungen mit den restlichen Feldern in der Natur. 123 Unser Feld φ hingegen wird isoliert betrachtet, wir nehmen an, dass es mit keinem anderen Feld wechselwirkt. Die krumme“ Funktion des Feldes heißt Diffeomorphismus, das ist eine Transforma- ” tion φ → ρ, die jedem Wert von φ einen Wert von ρ (Rho) zuordnet. Das bedeutet, ρ ist ebenfalls ein Quantenfeld, das auf eine bestimmte Weise vom Originalfeld φ abh¨angt. Die genaue Form dieser Abh¨angigkeit ist durch die Parameter a1, a2, a3,... des Diffeo- morphismus gegeben. In dieser Masterarbeit untersuchen wir, welche Eigenschaften das Feld ρ hat. Im Kapitel2 nehmen wir an, dass φ ein freies Feld ist. Wie erwartet stellt sich heraus, dass das Feld ρ nicht mehr als freies Feld erkennbar ist. Mathematisch ¨außert sich das dadurch, dass es nun Wechselwirkungsvertices (Gleichung 2.15) gibt. Allerdings sind diese Vertices zun¨achst nur eine mathematische Konstruktion. Was man - wenn es das Feld ρ in der Natur g¨abe - in einem Experiment beobachten wurde,¨ sind bestimmte n- Punkt Funktionen (n¨amlich die, fur¨ die die beteiligten Impulse onshell sind). Mit einer relativ komplizierten Rechnung kann man beweisen, dass die Vertices des Feldes ρ sich immer derartig gegeneinander aufheben, dass die n-Punkt-Funktionen gleich sind wie fur¨ das zu Grunde liegende Feld φ, das ist Theorem 2.2. Dieses Ergebnis war in der Literatur schon vorher bekannt. Es bedeutet, wenn man m¨ochte, kann man mit einem transformierten Feld ρ anstelle von φ rechnen; man wird dennoch dieselben Vorhersagen fur¨ physikalisch messbare Werte erhalten, lediglich die Zwischenschritte der Rechnung unterscheiden sich. Die folgenden beiden Kapitel3 und4 behandeln den Fall, dass das Feld φ selbst schon eine Wechselwirkung hat. Kapitel3 konzentriert sich auf die einfachste m ¨ogliche Form einer Wechselwirkung, Kapitel4 verallgemeinert diese Behandlung auf eine große Klasse m¨oglicher Wechselwirkungen. Wir k¨onnen zeigen (Theoreme 3.1 und 4.3), dass sich auch in diesem Fall die messbaren n-Punkt-Funktionen nicht ¨andern, wenn man einen Diffeomorphismus anwendet und das Feld ρ anstelle von φ betrachtet. Als naheliegende Anwendung“ dieser Wahlfreiheit widmen wir uns dem eingangs ” angedeuteten umgekehrten Problem: Wenn das Feld φ selbst schon eine Wechselwir- kung hat, gibt es dann einen Diffeomorphisms, sodass das transformierte Feld ρ wie ein freies Feld erscheint? Da wir wissen, dass die onshell n-Punkt-Funktionen sich bei Diffeomorphismen nicht ¨andern, betrachten wir eine 2-Punkt-Funktion, deren Impulse nicht onshell sind. In Theorem 4.2 zeigen wir, welchen Diffeomorphismus man w¨ahlen muss, damit diese Funktion genauso ist wie die entsprechende Funktion eines freien Skalarfeldes. Das so definierte Feld ρ nennen wir adiabatischen Diffeomorphismus. Im letzten Kapitel,5, untersuchen wir die Eigenschaften des adiabatischen Diffeomor- phismus ρ. Insbesondere ist ρ kein insgesamt freies Feld, sondern seine 2-Punkt-Funktion entspricht der eines freien Feldes nur fur¨ einen ganz bestimmten Impuls p. Die Ergebnisse dieser Masterarbeit haben keinerlei technische Anwendung, schon al- leine deshalb, weil das Feld φ in der Natur nicht existiert. Allerdings gibt es zwei große Probleme in der heutigen Quantenfeldtheorie, die formal ¨ahnlich zu den rechnerischen Effekten sind, die auftreten, wenn man φ durch einen Diffeomorphismus ρ ersetzt. Das erste ist eine konzeptionelle Schwierigkeit bei der Definition des Wechselwirkungsbildes in der Quantenfeldtheorie, siehe Abschnitt 1.5.1. Man nimmt dort an, dass auf eine be- stimmte Weise wechselwirkende Felder mit freien Feldern zusammenh¨angen, allerdings steht diese Annahme im Widerspruch zu Haag’s Theorem (1.2). Da auch die in dieser 124 Arbeit behandelten Diffeomorphismen einen Zusammenhang zwischen zwei (scheinbar verschiedenen) Feldern vermitteln, besteht die Hoffnung, dass man mit Hilfe von Diffeo- morphismen die Annahmen des Wechselwirkungsbildes mathematisch widerspruchsfrei formulieren k¨onnte. Das zweite Problem ist die bis heute unverstandene Theorie der Quantengravitation. Sie leidet stark vereinfacht unter der Tatsache, dass sie unendlich viele Wechselwir- kungsvertices enth¨alt, aber das trifft auch auf Diffeomorphismen ρ zu. Andererseits wissen wir, dass Diffeomorphismen die physikalisch messbaren Gr¨oßen nicht ver¨andern, das heißt, im Prinzip stellt auch ein Diffeomorphismus ρ eine sinnvolle Theorie dar, wenn die zu Grunde liegende Theorie φ sinnvoll war. Daher sind Diffeomorphismen ein Beispiel eines Quantenfeldes, das zwar unendlich viele Wechselwirkungsvertices enth¨alt, aber trotzdem physikalischen Sinn haben kann. Man k¨onnte prufen,¨ ob ein ¨ahnlicher Fall auch in der Quantengravitation vorliegt, ob es also eine zugrunde liegende Theorie gibt, die mathematisch weniger problematisch ist. Beide Probleme liegen weit jenseits dessen, was in der vorliegenden Arbeit uber¨ Diffeo- morphismen gesagt wird, und erfordern die Kl¨arung zahlreicher weiterer Fragen. Es ist dabei durchaus unsicher, ob letztendlich auch nur eines der beiden Probleme uberhaupt¨ mit Hilfe von Diffeomorphismen gel¨ost werden kann. Aber die bloße Tatsache, dass diese Probleme zumindest zu ¨ahnlichen Fragestellungen fuhren,¨ wie sie auch bei Diffeomor- phismen auftreten, dient als Motivation dafur,¨ sich mit letzteren zu besch¨aftigen. 125 C Danksagung Zwei verschiedene Dinge verdanke ich Dirk Kreimer: Erstens das Thema der Master- arbeit, das fur¨ mich im Laufe der Bearbeitung immer interessanter wurde und auch noch fur¨ sp¨ater einige Forschungsfragen aufwirft. Uber¨ das letzte Jahr hinweg hat er bereitwillig meine Arbeitsweise akzeptiert - dass ich wochenlang irgendetwas mache, ohne zwischendurch zu berichten, was eigentlich. Und zweitens drei Vorlesungen uber¨ fortgeschrittene Quantenfeldtheorie, die mir uberhaupt¨ gezeigt haben, dass es lohnend ist, sich heute noch“ mit der QFT zu besch¨aftigen, wo selbst B¨ackereiverk¨aufer von ” Stringtheorie reden, wenn sie mitbekommen, dass man Physiker ist. Dann danke ich meinen Eltern, die ohne Beschwerde hinnehmen, dass ich volle sie- ben Jahre lang studiere, mich die ganze Zeit unterstutzen¨ und schließlich auch diese Masterarbeit Korrektur gelesen haben. Auch Daniel hat Korrektur gelesen und mich konsequent auf unpr¨azise Formulierun- gen, uberfl¨ ussige¨ Halbs¨atze und fehlende Zitationen hingewiesen und darauf bestanden, dass ich wenigstens einmal einen konkreten experimentellen Zahlenwert zitiere. Das habe ich nur widerstrebend getan, denn wenn im mathematischen Verfahren das Un- ” bekannte zum Unbekannten einer Gleichung wird, ist es damit zum Altbekannten ge- stempelt, ehe noch ein Wert eingesetzt ist. Natur ist, vor und nach der Quantentheorie, das mathematisch zu Erfassende“ [HA88]. Zu guter Letzt verdanke ich Cordula zahlreiche Empfehlungen zu Layout und Ge- staltung, die dazu beigetragen haben, dass der Leser sich nur noch mit inhaltlichen Schwierigkeiten herumschlagen muss, aber nicht mehr mit der Struktur des Textes. 127 D Selbstst¨andigkeitserkl¨arung Hiermit erkl¨are ich, dass ich die vorliegende Masterarbeit selbstst¨andig und nur un- ter Verwendung der angegebenen Literatur angefertigt habe. Die aus fremden Quellen direkt oder indirekt ubernommenen¨ Stellen sind als solche kenntlich gemacht. Die Arbeit wurde bisher in gleicher oder ¨ahnlicher Form keiner anderen Prufungsbeh¨ ¨orde vorgelegt. Adlershof, 2. Juli 2018 129