p4 preliminary, August 21, 2008

Super-graphs with exponential coupling

Michael Karowski^{∗} Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

July 1971

Abstract:

Amplitudes in a Lagrangian quantum field theory with exponential coupling for arbitrary mass are defined in any order of perturbation theory. The regularization is unique by means of unitarity, causality and minimal singularity.

Contents

1 Introduction 1

2 “Super graphs” 2

3 Unitarity and causality 5

4 Minimal singularity 5

1 Introduction

In a previous paper [1] the superpropagator in a Lagrangian field theory with exponential coupling was investigated. This superpropagator is the vacuum expectation value of two exponential fields. In the present paper the results are generalized to any finite order in L_{I}. It is a well known fact that non-renormalizable theories are defined up to an infinite number of arbitrary constants. By means of unitarity and causality [2] the n-point Green’s function is determined by the m-point Green’s functions for m < n up to a distribution with support at x_{1} = ⋯ = x_{n}, i.e. in momentum space an entire function of order < 1/2 in p_{i}p_{j}. Thus the main problem is to give a \underline{unique} definition of these time ordered Green’s functions and to give physical plausible arguments for the special choice.

∗E-mail: karowski@physik.fu-berlin.de

For a class of non-polynomial couplings there exists a set of τ-functions which are in a certain manner “minimal singular” [3, 1]. it corresponds to the “simplest” theory belonging to a given Lagrangian [3]. The principal of “minimal singularity” is applicable to localizable^{1} theories with Lagrangians which are certain entire functions of the field.

We restrict ourselves to exp (f A(x)). More general cases can be generated by superposing exponentials with respect to f. In this paper the principle of “minimal singularity” is generalized to higher orders. It is shown that there exists a definition of the time ordered product

T : e^{fA(x_{1})} : ⋯: e^{fA(x_{n})} :

such that its vacuum expectation value, the n-point τ-function, is minimal singular at x_{1} = ⋯ = x_{n}. The function τ̃(p_{1}, …, p_{n}) decreases with p_{i}^{2} → −∞ for λ = −f^{2}/(4π^{2}) > 0 if all partial sums p_{i_{1}} +⋯+p_{i_{k}} are space like. As the arbitrariness of the n-point function is an entire function of order < 1/2 and these functions increase in all directions, the condition of “minimal singularity” implicates the uniqueness of the n-point τ-function. The case λ < 0 is more complicated and is not considered in this paper.

2 “Super graphs”

In this section we define the formal time-ordered product

T : e^{fA(x_{1})} : ⋯: e^{fA(x_{n})} : = e^{if^{2} ∑_{i<j} Δ_{F} (x_{i}−x_{j})} : e^{fA(x_{1})} … e^{fA(x_{n})} :  (1)

where A(x) is a free scalar field of mass m. The n-point function

τ(x_{1}, …, x_{n}) = e^{if^{2} ∑_{i<j} Δ_{F} (x_{i}−x_{j})}  (2)

correspond to a “super graph” with n vertices which are connected by super propagators. It is determined by the lower order τ-functions except at x_{1} = ⋯ = x_{n}.

The case n = 2 was discussed in a previous paper [1]. From the condition of minimal singularity follows uniquely the super propagator^{2}

iE_{F} (λ, x) + 1 = e^{if^{2}Δ_{F} (x)}  (3)

= \frac{1}{(2πi)^{2}} (∫_{C_{0}} dz ∫_{L_{0}} du + ∫_{C_{0}} dz ∫_{L_{z}} du) Γ(−u)λ^{u}g(u, z) (−x^{2} + io)^{−z}

1Localizable means: the fields are Jaffe fields [4], i.e. the Wightman functions in momentum space don’t increase stronger than an entire function of p^{2} of order < 1/2.

2The number 1 is represented by the residue at u, z = 0.

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where

λ = −f^{2}/(4π^{2}) > 0  (4)
g(u, z) = ∫_{0}^{∞} dtt^{z−1} (\frac{m K_{1}(m\sqrt{t})}{\sqrt{t}})^{u} .

The integration contour L_{z} encloses the ray parallel to the real axis from z to +∞ clockwise. by opening L_{z} to a line parallel to the imaginary axis we get C_{z}.

From (2) and (3) we get formally

τ(x_{1}, …, x_{n}) = ∏_{i<j} \{\frac{1}{(2πi)^{2}} ∫ dz_{ij} ∫ du_{ij}Γ(−u_{ij})λ^{u_{ij}}g(u_{ij}, z_{ij})\} f(x, z)  (5)

with

f(x, z) = ∏_{i<j} (− (x_{i} − x_{j})^{2} + io)^{−z_{ij}}

= ∏_{l=1}^{L} (− (x_{i_{i}} − x_{f_{l}})^{2} + io)^{−z_{l}}  (6)

where L = n(n − 1)/2 is the number of lines and N = (n − 1)(n − 2)/2 is the number of loops.

The product (6) was investigated by Speer [5] in the context of analytic renormalization. Let q_{1}, …, q_{L} and k_{1}, …, k_{N} be the momenta of the lines and loops, respectively. Then in momentum space

f̃(p, z) = δ (∑p_{j}) ∫ dk_{1} … dk_{N} ∏_{l=1}^{L} \{(−iπ^{2})\frac{Γ(2 − z_{l})}{Γ(z_{l})} (\frac{−q_{l}^{2} − io}{4})^{z_{l}−2}\}  (7)

with

q_{l} = ∑_{i=1}^{N} a_{li}k_{i} + ∑_{j=1}^{n} b_{lj}p_{j} , a_{li}, b_{lj} = 0, ±1 .  (8)

Doing the k-integrations we get for certain z_{l} (e.g. Re z_{l} < 0)

f̃(p, z) = δ (∑p_{j}) (−iπ^{2})^{L+N} ∏_{l=1}^{L} \{\frac{e^{iπ(2−z_{l})/2}}{Γ(z_{l})4^{z_{l}−2}} ∫_{0}^{∞} dα_{l}\} C^{−2}(α)e^{i(D(α,p)/C(α)+iϵ ∑ α_{l})} (9)

= δ (∑p_{j}) (−iπ^{2})^{L+N}Γ(−Z) \{∏_{l=1}^{L} \frac{1}{Γ(z_{l})4^{z_{l}−2}} ∫_{0}^{∞} dα_{l}α_{l}^{1−z_{l}}\}  (10)

× δ (1 − ∑α_{l}) C^{−2}(α) (\frac{−D(α, p)}{C(α)})^{Z}

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where

\frac{D(α, p)}{C(α)} = ∑_{l,l′} P_{l}P_{l′} (δ_{ll′} − ∑_{i,i′} Θ_{ii′}^{−1}α_{l}a_{li}a_{l′i′}) ,

Θ_{ii′} = ∑_{l=1}^{L} α_{l}a_{li}a_{l′i′} , P_{l} = ∑_{j=1}^{n} b_{lj}p_{j} , C(α) = det Θ_{j} , Z = ∑_{l=1}^{L} (z_{l} − 2) + 2N . (11)

Speer [5] proves that f(x, z) is a meromorphic function of z_{l} and

f(x, z) {∏_{A_{L}} Γ (∑_{l∈A_{l}} (z_{l} − 2) − 2N)}^{−1}  (12)

is holomorphic in ℂ^{L}. (The product is taken over all subsets A_{L} of {1, …, L}. There are more poles than those of the factor f(x, z) in (6). Hence the integration contours in (5) must be investigated carefully.

To get right prescription for the z-integrations in (5) we consider the case λ positive imaginary. Then the z-integrations may be taken along C_{0}

τ(x) = \frac{1}{(2πi)^{2L}} ∑_{k=0}^{L} ∑_{π}′ ∏_{l=1}^{k} \{∫ dz_{π(l)} ∫ du_{π(l)}\} ∏_{l=k+1}^{L} \{∫ dz_{π(l)} ∫ du_{π(l)}\} (13)

× ∏_{l=1}^{L} λ^{u_{l}}Γ(−u_{l})g(u_{l}, z_{l})f(x, z)

∑′_{π} means summation over all permutations with the condition

π(1) < … π(k) , π(k + 1) < … π(L).

We now shift the z_{π(l)}-integrations path for l < k, successively beginning with the last one, from C_{0} to loops L, paying attention to all poles of f(x, z). The z_{π(l)}-integrations path (l < k) must be replaced by the sum of loops

C_{0} → ∑_{A^{l}} L_{{∑_{l′∈π(A^{l−1})}}} .  (14)

The sum is taken over all sunsets A^{l} of {1, …, l} containing l. After this shifting of integration paths, (13) converges for all λ with |arg λ| ≤ π. The arguments for the convergence at z → ∞ are similar to those in [1]. For λ > 0 we define

τ(x) = \frac{1}{(2πi)^{2L}} ∫ dz ∫ du ∏_{l=1}^{L} \{λ^{u_{l}}Γ(−u_{l})g(u_{l}, z_{l})\} f(x, z) .  (15)

The integrations have to be taken according to (13) and (14).

(The limit ϵ → 0 may produce infrared singularities. However, these cancel at last in a theory with m ≠ 0. To avoid this difficulty one may use, instead of (3) another, more complicated, representation [1].)

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3 Unitarity and causality

A set of time ordered products of fields has to satisfy the following conditions [2]: Unitarity:

∑_{k=0}^{n}(−1)^{k} ∑_{σ}′T^{†} (B(x_{σ(1)}) … B(x_{σ(k)})) T (B(x_{σ(k+1)}) … B(x_{σ(n)})) = 0  (16)

Causality:

∑_{k=0}^{n}(−1)^{k} ∑_{σ}′T^{†} (B(x)B(x_{σ(1)}) … B(x_{σ(k)})) T (B(x_{σ(k+1)}) … B(x_{σ(n)}))

= i^{n}R(x; x_{1}, ….n_{n}) = 0 , if one x_{i}^{0} < x^{0} .  (17)

(T^{†} is the anti-time ordered product and ∑′_{σ} as in (13).) These conditions follow immediately from the meaning of time ordering.

For B(x) = : exp (fA(x)) : the unitarity condition is

∑_{k=0}^{n}(−1)^{k} ∑_{σ}′e^{if^{2} ∑_{i<j} Δ_{ijk}(x_{σ(i)}−x_{σ(j)})} : e^{fA(x_{1})} … e^{fA(x_{n})} : = 0  (18)

with

Δ_{ijk} = \{ \begin{matrix}−Δ^{∗} & for & i, j ≤ k \\ Δ_{+} & for & i ≤ k < j \\ Δ_{F} & for & k < i, j . \end{matrix}  (19)

We now show that the coefficients of the normal ordered product in (18) vanishes: The following cutting formula [6] holds foe certain z_{ij} (e.g. Re z_{ij} < 0)

∑_{k=0}^{n}(−1)^{k} ∑_{σ}′ ∏_{i<j} (− (x_{σ(i)} − x_{σ(j)})^{2} + ioδ_{ijk})^{−z_{ij}} = 0  (20)

with

δ_{ijk} = \{ \begin{matrix}−1 & for & i, j ≤ k \\ x_{σ(i)}^{0} − x_{σ(j)}^{0} & for & i ≤ k < j \\ 1 & for & k < i, j . \end{matrix}

The terms in (20) have at most the singularities as f(x, z). Therefore the coefficient in (18) has the representation (15) and vanishes because of (20). Similarly causality may be proved.

4 Minimal singularity

In this section we investigate the asymptotic behaviour of the n-point τ-functions in momentum space. We want to show that τ̃(p_{1}, …, p_{n}) decreases with p_{i}^{2} → −∞ for λ > 0 if all partial sums p_{i_{1}}+⋯+p_{i_{k}} are space like. This property is called “minimal singularity”, because the decreasing means that τ(x_{1}, …, x_{n}) has no singularities at x_{1} = ⋯ = x_{n}. More precisely (for λ > 0)

p4 preliminary, August 21, 2008

⟨τ, φ⟩ = lim_{ρ→∞}⟨τ, χ_{ρ}φ⟩ for all φ ∈ ℭ  (21)

with χ_{ρ} ∈ ℭ_{M} , χ_{ρ} bounded, χ_{ρ}(x⃗_{i}) ≡ 0 in a neighbourhood of x⃗_{1} = ⋯ = x⃗_{n} and χ_{ρ}(x⃗_{i}) ≡ 1 if one |x⃗_{i} − x⃗_{j}| > ρ. The τ-functions are defined uniquely by unitarity, causality and minimal singularity. This uniqueness can also be see in momentum space. The npoint τ-function is defined by the m-point τ-functions for m < n by means of unitarity and causality up to a distribution with support at x_{1} = ⋯ = x_{n} or in momentum space up to an entire function of order < 1/2 in p_{i}p_{j}, and this function does not decrease in any direction.

For λ > 0 the decreasing property can easily be shown. If all partial sums p_{j_{1}} +⋯+p_{j_{n}} are space-like, −D(α, p)/C(α) is positive [7]. Thus the z-integration in (15), transferred to momentum space, converge along C_{0} (like in (13)). The asymptotic behaviour is determined by the nearest singularity on the left of the integration path. But there is no singularity. Hence τ̃(p_{1}, …, p_{n}) decreases stronger than any power with p_{j}^{2} → −∞.

For a special case we derive a more explicit formula for the asymptotic behaviour: Consider

τ̃(p_{1}, p_{2}, 0, …, 0) = δ(p_{1} + p_{2})τ̃_{n}(p)

which corresponds to the n-th order of the two-point function. (−D(α, p)/C(α))^{Z} can be written as (−p^{2})^{Z}α_{1}^{Z}(1 − α_{1}β)^{Z} with β = ∑_{i,i′} θ_{ii′}^{−1}a_{1i}a_{1i′} and 0 ≤ α_{1}β ≤ 1.

To get the asymptotic behaviour, we have to substitute [1].

G(λ, z) = \frac{1}{2πi} [∫_{L_{1}} du + ∫_{L_{z}} du] Γ(−u)λ^{u}g(u, z)
→ cλ^{z−λa}Γ(z − λa) , λ > 0

with c = e^{−λa(ln aγ−1)}, a = m^{2}/4 and γ = Euler’s constant. Thus

τ̃_{n}(p) ∼ ∏_{l=1}^{L} \{\frac{1}{2πi} ∫ dz_{l} \frac{Γ(λa − z_{l})}{Γ(z_{l})} ∫_{0}^{1} dα_{l} α_{l}^{1−z_{l}}\} δ (1 − ∑α_{l}) C^{−2}(α)

× (−λp^{2}/4)^{Z} ∑_{r=0}^{∞} \frac{1}{r!}Γ(r − Z)β^{r}α_{1}^{Z+r}

substituting z_{1}, …, z_{L} → Z, …, z_{L} and integrating over Z we get

τ̃_{n}(p) ∼ ∑_{r=0}^{∞} \frac{1}{r!} ∏_{l=2}^{L} \{\frac{1}{2πi} ∫ dz_{l} \frac{Γ(λa − z_{l})}{Γ(z_{l})} ∫_{0}^{1} dα_{l} α_{l}^{1−z_{l}}\} ∫_{0}^{1} dα_{1} α_{1}^{Z′+r−1}
× β^{r}δ (1 − ∑α_{l}) C^{−2}(α)G_{03}^{20}(−λp^{2}/4|λa + Z′ − 2, r, Z′ − 1)

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REFERENCES 7

where Z′ = ∑_{l=2}^{L}(z_{l} − 2) + ”N and G_{03}^{20} is Meijer’s function. The leading term for p^{2} → ∞ is determined by the first singularity in Z′. This is a pole at Z′ = −r, which comes from the α_{1}-integration at α_{1} = 0. Thus the asymptotic behaviour is

τ̃_{n}(p) ∼ G_{03}^{20}(−λp^{2}/4|λa′ − 2, 0, −1)

This is independent of n (up to a constant factor which may depend on n). Hence it seems to be not possible that there is any cancellation in the perturbation expansion with respect to to bad high-energy behaviour.

References

[1] M. Karowski, Commun. Math. Phys., 19 (1970) 289 and DESY preprint 70/20 (1970).

[2] N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields, Intersience Publishers, Ins., New York 1959.

[3] H. Lehmann and K. Pohlmeyer, Commun. Math. Phys. 20 (1971) 101.

[4] A. Jaffe, Phys. Rev. Lett. 17 (1966) 661, Phys. Rev. 158 (1967) 1454.

[5] E. Speer, J. Math. Phys. 9 (1968) 1404.

[6] M. Veltman, Physica 29 (1963) 186.

[7] K. Symanzik, Progr. Theor. Phys. 20 (1958) 690.

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