can somebody explain or point to the relating mathematics showing Why coupling constants with negative mass dimensions lead to non-renormalizable theories?
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The standard argument goes as follows. For a connected Feynman diagram$^1$ the superficial degree of (UV) divergence $D$ is equal to$^2$ $$\begin{align} D~:=~& \#\{\text{$\mathrm{d}p$ in int. measure}\} ~+~ \#\{\text{$p$ in numerator}\}\cr &~-~ \#\{\text{$p$ in denominator}\}\cr\cr ~=~& Ld +\sum_i V_i d_i + \sum_f [\widetilde{G}_{0f}]I_f\cr ~\stackrel{\text{Ref. }3}{=}& \left(\sum_f I_f -(\sum_i V_i -1)\right)d \cr & +\sum_i V_i d_i + \sum_f(2[\phi_f]-d) I_f \cr ~=~&d- \sum_i(d-d_i) V_i + \sum_f[\phi_f] ~2I_f \cr ~=~&d- \sum_i(d-d_i) V_i +\sum_f[\phi_f] \left(\sum_i V_i n_{if}-E_f\right) \cr ~=~& d - \sum_i \left(d - d_i - \sum_f [\phi_f] n_{if}\right) V_i - \sum_f [\phi_f] E_f\cr ~=~& d - \sum_f [\phi_f] E_f - \sum_i [\lambda_i] V_i \cr ~\stackrel{\text{Ref. }4}{=}& [\text{amputated diagram}] - \sum_i [\lambda_i] V_i, \end{align}\tag{1} $$ where
$d$ is the number of spacetime dimensions;
$[\cdot]$ denotes the mass dimension in units where $\hbar=1=c$;
$L$ is the number of independent loops;
$I_f$ is the number of internal lines with a free propagator $\widetilde{G}_{0f}$ in the Fourier momentum space of a field $\phi_f$ of type $f$;
$V_i$ is the number of vertices of $i$'th interaction type with coupling constant $\lambda_i$, $d_i$ number of spacetime derivatives, and $n_{if}$ legs of type $f$;
$E_f$ is the number of amputated external lines with a field $\phi_f$ of type $f$.
The formula (1) has in principle a simple interpretation in terms of double-entry bookkeeping as follows. Recall that each vertex arises from a dimensionless action term. So instead of debiting the loop momentum variables $p$ [cf. the definition of $D$], we can instead credit [with the opposite sign] the mass dimension of the rest of the Feynman diagram, namely coupling constants and amputated legs [cf. formula (1)].
Let us now return to OP's question. If an interaction vertex, say of type $i_0$, has $[\lambda_{i_0}]<0$, then eq. (1) indicates that we can build infinitely many superficially divergent Feynman diagrams with $D\geq 0$ by using more and more vertices of type $i_0$. This render the theory non-renormalizable in the old Dyson sense.
References:
S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; eq. (12.1.8).
M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; eqs. (10.11) + (10.13).
Use my example to explain why loop diagram will not occur in classical equation of motion?
Why do all Feynman diagrams with same number of external legs have the same mass dimension?
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$^1$ We assume that the sources $J_k$ are either stripped from the Feynman diagram or are delta functions in momentum space so that the external legs carry fixed 4-momenta.
$^2$ It is implicitly assumed that the coefficients in front of the kinetic terms in the action are dimensionless. The quantity $[\phi_f]$ is non-negative for $d\geq 2$.
Qmechanic
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Notes for later: 1. Q: Can we generalize to not putting $\hbar=1$? A: $[\cdot]$ then denotes inverse length dimension. Momentum $p$ should be replaced with wavevector $k$. And $[\phi_f]$ should be replaced with $[\frac{\phi_f}{\sqrt{\hbar}}]$. And $[\lambda_i]$ should be replaced with $[\hbar^{n_i/2-1}\lambda_i]$, where $n_i=\sum_f n_{if}$ is the number of legs on a vertex of $i$th type. 2. Q: How about not putting $c=1$ for non-relativistic physics? – Qmechanic Feb 07 '22 at 12:15
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Notes for later: https://userswww.pd.infn.it/%7Eferuglio/Falkowski.pdf p.6: eq. (1.7) blocking/dimless coupling constants. p. 6-7: $\hbar$. – Qmechanic May 22 '22 at 20:19
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Notes for later: 1. For a disconnected diagram with $\pi_0$ connected components, the first term $d\to \pi_0d$ is replaced in eq. (1). 2. In QM $d=1$ the scalar field $[\phi]=\frac{d-2}{2}=-\frac{1}{2}$, so $D$ increased with number of external legs. – Qmechanic Jul 06 '22 at 11:31
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Notes for later: $\quad \sum_i V_i ( n_i -2) =2(L-1) +\sum_f E_f $. So at a certain $\hbar$/loop-order, the number of vertices $V_i$ with $n_i\geq 3$ legs (in the amputated correlation function) is finite. 2-vertices (apart from the kinetic terms) contain derivatives. If we assume Lorentz symmetry such 2-vertices are irrelevant $[\lambda_i]<0$. If the kinetic term is unique (e.g. by homogeneity & isotropy), then all other 2-vertices has $[\lambda_i]\neq 0$. It is natural to assume that all relevant or marginal 2-vertices belong to the free (rather than the interaction) part of the action. – Qmechanic Jun 15 '23 at 07:33
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- E. Panzer, PhD thesis, https://arxiv.org/abs/1506.07243 Be aware that Panzer's SDOD $\omega=-\frac{D}{2}$ in eq. (2.1.18) is normalized with a factor $-\frac{1}{2}$. Laplacian $(V!-!\pi_0)\times(V!-!\pi_0)$ matrix. 6. https://arxiv.org/abs/0910.0114 Lemma 50. 7. https://arxiv.org/abs/0910.5429 8. Bogner & Weinzierl https://arxiv.org/abs/1002.3458 Laplacian $V\times V$ matrix. 9. https://arxiv.org/abs/2202.12296 $(+,-,-,-)$. $\rm L$ seems to be missing in eq. (12). Eq. (B1) correct. 10. https://www.osti.gov/etdeweb/biblio/21380826 15. https://math.stackexchange.com/a/4764382/11127
– Qmechanic Aug 17 '23 at 08:51 -
$n$-tree=tree with $n$ conn. comp. Spanning subgraph=touches all vertices. (Touch by vertex is allowed, and is in fact assumed to be the case.) $(n,\ell)$-graph=graph with $n$ conn. comp. and $\ell$ loops. Componentwise 1PI=don't cut into more conn. comp. by removing 1 line. Componentwise 1VI=don't cut into more conn. comp. by removing 1 vertex. (I.e. the notions make sense even if the graph is not connected.) A self-loop $\int\frac{d^dk}{\pi^{d/2}}\frac{\hbar}{i}\frac{\Gamma(\nu)}{(k^2+m^2-i\epsilon)^{\nu}}=\hbar m^{d-2\nu}\Gamma(\nu!-!\frac{d}{2})$ factorizes. Assume no self-loops. – Qmechanic Aug 27 '23 at 11:32
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Amputated not nec. conn. graphs, i.e. each end of a line has a vertex. (Paradigm: Trees are graphs.) SDOD $\omega(G)\equiv D(G)=dL(G)-2\sum_{e\in G}\nu_e$. Future notation: $G\equiv\Gamma$. $\quad k_v=\sum_{e\in E}k_e{\cal E}{e,v}+p_v$; $\quad{\cal L}^{\prime}:={\cal E}^{\prime T}{\cal A}^{-1}{\cal E}^{\prime}$; $\quad{\cal U}{0,0}\equiv{\cal U}0\equiv{\cal U}={\cal K}\det{\cal A}$; $\quad{\cal K}{0,0}\equiv{\cal K}_0\equiv{\cal K}=\det{\cal L}^{\prime}$; $\quad{\cal F}/{\cal U}={\cal F}_0/{\cal U} +m^T{\cal A}m$; $\quad{\cal F}_0/{\cal U}=p^{\prime T}{\cal L}^{\prime -1}p^{\prime}$; – Qmechanic Aug 29 '23 at 08:23
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$\quad{\cal F}0/\det{\cal A}=p^{\prime T}\text{Cofactor}({\cal L}^{\prime})p^{\prime}$ $=\sum^{\prime}{i,j\in V\backslash{v_0}}p_i(-1)^{i+j}\det({\cal L}^{\prime}{}^i_j)p_j$; Homogeneity:$\quad[{\cal U}{n,\ell}/\det{\cal A}]=[{\cal K}{n,\ell}]=[\alpha^{-1}]^{V-\pi_0-n+\ell}$; $\quad[{\cal U}{n,\ell}]=[\alpha]^{L+n-\ell}$; $\quad[{\cal F}{n,\ell}/\det{\cal A}]=[\alpha^{-1}]^{V-\pi_0-1-n+\ell}$; $\quad[{\cal F}{n,\ell}/{\cal U}{n,\ell}]=[\alpha]$; $\quad[{\cal F}_{n,\ell}]=[\alpha]^{L+1+n-\ell}$; $\quad [\alpha]=\text{mass dim}^{-2}$; – Qmechanic Aug 29 '23 at 10:09
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Multiple parallel edges (bananas) are allowed. $\quad\frac{\Gamma(\nu)}{k^2+m^2-i\epsilon}=i^{\nu}\int_{\mathbb{R}_+}d\alpha~\alpha^{\nu-1}e^{-i(k^2+m^2-i\epsilon)\alpha}$; Schwinger parametrization (without coupling constant factor $\prod_v\tilde{g}_v $ and traditional physics Fourier factor $(4\pi)^{-dL/2}$). Minkowski signature $(-,+,+,+)$. https://physics.stackexchange.com/q/752398/2451 – Qmechanic Aug 29 '23 at 14:24
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$\quad I=\left[\prod_{e\in E}\frac{\hbar}{i}\int\frac{d^dk_e}{\pi^{d/2}}\frac{\Gamma(\nu_e)}{(k_e^2+m_e^2-i\epsilon)^{\nu_e}}\right]\left[\prod^{\prime}{v\in V\backslash{v_0}}\frac{i}{\hbar}\pi^{d/2}\delta^d(k_v)\right]$ $=\left[ \prod{e\in E} \frac{\hbar}{i}i^{\nu_e}\int_{\mathbb{R}+}d\alpha_e~\alpha_e^{\nu_e-1}\int\frac{d^dk_e}{\pi^{d/2}}\right]\left[\prod^{\prime}{v\in V\backslash{v_0}}\frac{i}{\hbar}\int\frac{d^dx_v}{(4\pi)^{d/2}}\right]\exp(-iS)$ – Qmechanic Aug 30 '23 at 15:26
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$I=\ldots=\left(\frac{\hbar}{i}\right)^Li^Li^{-dL/2}\left[\prod_{e\in E}i^{\nu_e}\int_{\mathbb{R}+}d\alpha_e~\alpha_e^{\nu_e-1}\right]{\cal U}^{-d/2}\exp(-i({\cal F}/{\cal U}-i\epsilon))$ non-projective $=\hbar^Li^{-D(G)/2}\left[\prod{e\in E}\int_{\mathbb{R}+}d\alpha_e~\alpha_e^{\nu_e-1}\right]\int{\mathbb{R}+}!d\lambda~\delta(\lambda!-!\sum{e\in E}H_e\alpha_e){\cal U}^{-d/2}\exp(-i({\cal F}/{\cal U}-i\epsilon))$ linear gauge. – Qmechanic Sep 04 '23 at 06:48
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$I=\ldots\stackrel{\alpha^{\prime}e=\lambda\alpha_e}{=}$ $\hbar^L\left[\prod{e\in E}\int_{\mathbb{R}+}d\alpha_e~\alpha_e^{\nu_e-1}\right]\delta(1!-!\sum{e\in E}H_e\alpha_e)\frac{\Gamma(-D(G)/2)}{{\cal U}^{d/2}}\left(\frac{\cal F}{\cal U}\right)^{D(G)/2}$ projective/scale invariant/Cheng-Wu theorem; (Compare with I&Z eq. (6-91) with $(+,-,-,-)$.) – Qmechanic Sep 06 '23 at 10:30
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$\quad S=\sum_e\alpha_e(k_e^2+m_e^2-i\epsilon) +\sum^{\prime}_{v\in V\backslash{v_0}}k_v\cdot x_v$ $=k^T{\cal A}k +x^{\prime T}({\cal E}^{\prime T}k+p^{\prime}) +m^T{\cal A}m-i\epsilon$ $=(k+{\cal A}^{-1}{\cal E}^{\prime}\frac{x^{\prime}}{2})^T{\cal A}(k+{\cal A}^{-1}{\cal E}^{\prime}\frac{x^{\prime}}{2}) -\frac{x^{\prime T}}{2}{\cal L}^{\prime}\frac{x^{\prime}}{2} +x^{\prime T}p^{\prime} +m^T{\cal A}m-i\epsilon$ – Qmechanic Sep 08 '23 at 12:35
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$S=\ldots=(k+{\cal A}^{-1}{\cal E}^{\prime}\frac{x^{\prime}}{2})^T{\cal A}(k+{\cal A}^{-1}{\cal E}^{\prime}\frac{x^{\prime}}{2})$ $-(\frac{x^{\prime}}{2}-{\cal L}^{\prime -1} p^{\prime})^T{\cal L}^{\prime}(\frac{x^{\prime}}{2}-{\cal L}^{\prime -1}p^{\prime}) +p^{\prime T}{\cal L}^{\prime -1}p^{\prime}+m^T{\cal A}m-i\epsilon$; For $L=1$ one-loop, choose gauge fixing fct. $\sum_{e\in E}H_e\alpha_e={\cal U}$. Note that kin. ${\cal F}$ drops out if $D(G)=0$ log. div. Period = Residue of $1/\epsilon$ in log. div. graph with no kin. $\quad P(G)={\rm Res}_{D(G)=0}I(G)=\int\ldots {\cal U}(G)^{-d/2}$; – Qmechanic Sep 09 '23 at 14:43
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Notes for later (old): $k_v=\sum_{e\in E}k_e{\cal E}{e,v}+\Delta p_v$; $\quad \Delta p_v:=p_v-q$; $\quad I=\left[\prod{e\in E}\int\frac{d^dk_e}{\pi^{d/2}} \frac{\Gamma(\nu_e)}{(k_e^2+m_e^2)^{\nu_e}}\right]\left[\prod^{\prime}{v\in V\backslash {v_0}} \pi^{d/2} \delta^d (k_v)\right] $ $ =\left[ \prod{e\in E} \int_{\mathbb{R}+}d\alpha_e~\alpha_e^{\nu_e-1}\int\frac{d^dk_e}{\pi^{d/2}} \right]\left[\prod{v\in V}\int\frac{d^dx_v}{(4\pi)^{d/2}} \right]\int\frac{d^dq}{(2\pi)^d} \exp(-\widetilde{S})$; $\quad {\cal L}:={\cal E}^T{\cal A}^{-1}{\cal E}$; – Qmechanic Sep 11 '23 at 07:33
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Notes for later (old): $\quad \widetilde{S}=\sum_e \alpha_e(k_e^2+m_e^2) + i\sum_{v\in V}k_v\cdot x_v =k^T{\cal A}k +i x^T({\cal E}^Tk+\Delta p) +m^T{\cal A}m$ $=(k+i{\cal A}^{-1}{\cal E}\frac{x}{2})^T{\cal A}(k+i{\cal A}^{-1}{\cal E}\frac{x}{2}) +\frac{x^T}{2}{\cal L}\frac{x}{2} +ix^T\Delta p +m^T{\cal A}m$; – Qmechanic Sep 12 '23 at 13:32
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$\quad{\cal L}{ii}=\sum{e\in i}\alpha_e^{-1}$; $\quad{\cal L}{i\neq j}=-\alpha_e^{-1}$ if $i\in e\ni j$ else $0$; $\quad {\cal E}\begin{pmatrix} 1\cr \vdots \cr 1 \end{pmatrix} =0$ $\quad\Rightarrow\quad$ $\det{\cal L}=0$; To show that $\det({\cal L}^1_1)=\det({\cal L}^n_n)$ (for conn. graph) compare ${\cal L}^1_1$, ${\cal L}^{1,n}{1,n}$, ${\cal L}^n_n$. Primary $e_0$-sector: $\alpha_e\leq\alpha_{e_0}$. Hepp sectors: $\alpha_{\pi(1)}\leq\ldots\leq\alpha_{\pi(I)}$. – Qmechanic Sep 13 '23 at 08:10
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Assume $D(G)<0$ is UV conv.?? UV-div $k_e\to\infty\Rightarrow\alpha_e\to0$. IR/soft/collinear div. $k_e,m_e,p_v\to0\Rightarrow\alpha_e\to\infty$. Tropical formulas $\quad\alpha_e\equiv e^{\tau_e}$; Tropical $\tau$-function (of graph $G$)$=\sum_{e\in G}\nu_e\tau_e-\frac{D(G)+d}{2}\max{\cal U}{00}(G)+\frac{D(G)}{2}\max{\cal F}{00}(G)$; Momentum-IBPs are often more practical than $\alpha$-sector-decomposition. – Qmechanic Sep 13 '23 at 08:10
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UV: Tropical $\alpha$-degree (of subgraph $\gamma$)$=\sum_{e\in\gamma}\nu_e\tau_e-\frac{D(G)+d}{2}L(\gamma)+\frac{D(G)}{2}(L(\gamma)(+1)0)$ $=-\frac{D(\gamma)}{2}\left(+\frac{D(G)}{2}\right)_0>0$ for UV conv (Gen. case $\neq 0$); IR: (of subgraph $G/\gamma$)$=-\frac{D(G/\gamma)}{2}\left(+\frac{D(G)}{2}\right){\neq0}<0$ for IR conv (0 case most important); Test ${\cal F}_{00}(G/\gamma)=0$; (Ignoring gauge-fixing. Panzer seems to argue via the non-projective formula.) Note that SDOD grows if we destroy edges without destroying loops. Hence it is enough to check 1PI subgraphs. – Qmechanic Sep 15 '23 at 07:10
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- Dirk Kreimer. UV $\overline{MS}$ Renormalization. https://arxiv.org/abs/q-alg/9607022 $<\ldots>=$div. part p.23. Knot/link p.30. Skein relation p.32. ${}_j\Delta$ p.33. $\zeta(3)$=trefoil p.68. https://arxiv.org/abs/q-alg/9707029 Hopf algebra. $Z=$CT. $\quad-R$=div. part. Forest formula is non-overlapping p.5. https://arxiv.org/abs/hep-th/9810022 Overlapping divergences (OD). https://arxiv.org/abs/1512.06409 Cosmic Galois group. Prop 2.2. Def 3.1 https://arxiv.org/abs/2003.04301
– Qmechanic Sep 15 '23 at 07:10 -
- Overlapping divergencies destroys the paradigm of only needing to renormalizing 1PI vertices and self-energies. Kreimer: "One of the achievements of renormalization theory that it disentangles ODs in terms of nested and disjoint ones." Idea: Does ODs go away in $\alpha$-parametrization?
– Qmechanic Sep 15 '23 at 12:13 -
Matrix-Tree Theorem: 20. https://en.wikipedia.org/wiki/Laplacian_matrix 21. https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem 22. https://en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula 23. https://julesjacobs.com/notes/kirchoff/kirchoff.pdf 24. https://www.math.brown.edu/reschwar/M1230/cauchy.pdf 25. https://personalpages.manchester.ac.uk/staff/mark.muldoon/Teaching/DiscreteMaths/LectureNotes/IntroToMatrixTree.pdf No proof but mentions Tutte. 26. https://mathoverflow.net/q/73385 Induction. – Qmechanic Sep 15 '23 at 14:07
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Lagrangian/determinants originally comes from Gaussian integration but polynomials such as Kirchhoff, Symanzik, Tutte, etc, seem to be the right generalization (incl. self-loops). 39. https://www.mathematicalgemstones.com/gemstones/a-linear-algebra-free-proof-of-the-matrix-tree-theorem/ Blog with Zeilberger's idea. – Qmechanic Sep 16 '23 at 13:22
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- https://sites.math.rutgers.edu/~zeilberg/mamarimY/DM85.pdf complete graph p.67. (Incompleting a graph can destroy but not create loops.) Each vertex in directed graph can only have 1 output (0 if we delete rows). Deleting $v$th row<=>$v\in I$<=>$v$th vertex terminal<=>$v$ has no output. Idea: Introduce $r_{vw}$ with $n=E<w\leq V=n+\pi_0$ for terminal vertices. (Recall that a single vertex is a 1-tree.) Case $I=J$: Each conn. comp. of a spanning tree must have precise 1 terminal vertex $v\in I$. OTOH, $w\in J\backslash I$ is never a terminal vertex, possibly an initial vertex.
– Qmechanic Sep 17 '23 at 02:13 -
- J.W. Moon https://doi.org/10.1016/0012-365X(92)00059-Z Digraph = directed graph. Their "root" is more correctly called terminal element. Thm 2.1: Spanning sub-graph=each dynamical (= not spectator) vertex $v\notin I$ has an output but not nec. an input, i.e. $-s/r_{v\Box}$ is present in each term. So $n$ edges. Each vertex has 0 or 1 output, possibly many inputs. Don't forget that cycles of length $\geq 3$ are directed.
– Qmechanic Sep 18 '23 at 07:26 -
$\quad\det(M)=\sum_{\pi\in S_n}\sigma(\pi)m_{1\pi(1)}m_{2\pi(2)}\ldots m_{n\pi(n)}$ $=\sum_{\pi\in S_n}\sigma(\pi)m_{\pi(1)\pi(2)}m_{\pi(2)\pi(3)}\ldots m_{\pi(n)\pi(1)}$; $s$-variables comes in cycles only. Each $s$-loop has a minus. Now consider the remaining diag. $r$-variables. Each conn. comp. has 0 or 1 loops. Consider spanning graph $D=B\cup\bigcup_1^{\nu(D)} C$. [Old:Make disjoint partition $R\sqcup S={1,\ldots,n}$ of rows.] $\quad\det(M)=\sum_{\text{directed graph }D}r(b(D))\prod_{h=1}^{\nu(D)}{r(c_h(D))-s(c_h(D))}$. Error: $\nu(D)=$# of cycles rather than conn. comp.?? – Qmechanic Sep 18 '23 at 08:09
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Thm 3.1 with $s=r$: The vertex corresponding to the $w$th deleted column must be hit. Spanning graph=each dynamical (=not spectator) vertex $v\notin I$ has an output but not nec. an input. $\quad w\in I\backslash J$ has input but no output. $\quad w\in I\cap J$ has no output; possibly input. $\quad w\in I$ can only sit in second place of $-s/r_{\Box w}$. $\quad v\in J\backslash I$ has output; possibly input. $\quad v\notin I\cup J$ has output; possibly input. (Ignoring conn. comp. consisting of a single vertex.) – Qmechanic Sep 18 '23 at 11:19
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$\det(M^I_I)=\sum_{D\text{ with terminal vertices }I}r(b(D))\prod_{h=1}^{\nu(D)}{r(c_h(D))-s(c_h(D))}$ $=\sum_{|I|\text{-tree }F\text{ with terminal vertices }I}s(e(F))$; $\quad\Gamma\backslash\gamma$ ($\Gamma/\gamma$) is $\Gamma$ where each edge of $\gamma$ is removed (shrunk to a point and its vertices identified), respectively. The quotient can have self-loops. "Double" quotient $\Gamma//\gamma$ is $\emptyset$ if $L(\gamma)>0$. Set deletion may violate mom. cons. (Paradigm: Try to avoid set deletion $\Gamma\backslash\gamma$.) Remove pertinent row/column in Lagrangian. – Qmechanic Sep 19 '23 at 09:36
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$\quad\det(0\times0\text{-matrix})=1$; A single vertex is a 1-tree. $\quad{\cal K}{n,\ell}(v)=\delta_n^0\delta{\ell}^0={\cal U}{n,\ell}(v)$; $\quad{\cal F}{n,\ell}(v)=0$; Undirected graphs: $\quad{\cal K}{0,0}(v_1!\ni!e!\in!v_2)=\alpha_e^{-1}$; $\quad{\cal U}{0,0}(v_1!\ni!e!\in!v_2)=1$; $\quad{\cal F}_{0,0}(v_1!\ni!e!\in!v_2)=p_1\cdot p_2 \alpha_e$; (even for self-loop $v_1=v_2$ if we change loop number $\ell=0\to\ell=1$.) – Qmechanic Sep 21 '23 at 06:24
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$\quad{\cal K}{n,\ell}(G)=\sum{\text{span. $(|\pi_0(G)|+n,\ell)$-subgraph }\gamma}\prod_{e\in\gamma}\alpha_e^{-1}$; $\quad{\cal U}{n,\ell}(G)=\sum{\text{span. $(|\pi_0(G)|+n,\ell)$-subgraph }\gamma}\prod_{e\notin\gamma}\alpha_e$; $\quad{\cal F}{n,\ell}(G)=\sum{\text{vertex pair }i,j}p_i\cdot p_j$ $\sum_{\text{span. $(|\pi_0(G)|+1+n,\ell)$-subgraph }\gamma\text{ with terminal vertices }i,j}\prod_{e\notin\gamma}\alpha_e$; $\quad n,\ell\in\mathbb{Z}$. (If $n,\ell<0$ then it is zero.) Laplacian factorizes in conn. comp. – Qmechanic Sep 21 '23 at 06:24
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Vanishing theorem: ${\cal K}{n,\ell}(G)=0\Rightarrow{\cal K}{\leq n,\geq\ell}(G)=0$. Similar for ${\cal U}{n,\ell}(G)$ and ${\cal F}{n,\ell}(G)$. If $G$ has no external momenta $p_v=0$ then ${\cal F}{n,\ell}(G)=0$. If $G$ has no masses, similar story. Scaleless case. $\quad {\cal U}{n=0,\ell}(\text{tree})=\delta_{\ell}^0$; – Qmechanic Sep 22 '23 at 10:17
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If $G_1\cap G_2\subseteq v$ then ${\cal K}n(G_1\cup G_2)=\sum{r=0}^n{\cal K}r(G_1){\cal K}{n-r}(G_2)$; $\quad{\cal U}n(G_1\cup G_2)=\sum{r=0}^n{\cal U}r(G_1){\cal U}{n-r}(G_2)$; $\quad{\cal F}n(G_1\cup G_2)=\sum{r=0}^n{\cal F}r(G_1){\cal U}{n-r}(G_2)+\sum_{r=0}^n{\cal U}r(G_1){\cal F}{n-r}(G_2)$; $\det({\cal L}^{\prime v}_w(G))=0$ if $v$ and $w$ are not connected in $G$. – Qmechanic Sep 22 '23 at 10:17
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Induction: $\quad{\cal K}{n,\ell}(G)=\theta(e!\in!G)\left(\alpha_e^{-1}{\cal K}{n,\ell}(G/e)+{\cal K}{n,\ell}(G\backslash e)\right)$ $+\theta(e!\notin!G)\left{\theta(v_1,v_2!\in!G)(\ldots{\cal K}{n-1,\ell}(G/e)+\ldots{\cal K}{n,\ell+1}(G/e)) +(1!-!\theta(v_1,v_2!\in!G)){\cal K}{n,\ell}(G\backslash e)\right}$ (spanning subgraph $\gamma\subseteq G$: cases $e\in\gamma$; $G\ni e\notin\gamma$; $e\notin\gamma,G$); – Qmechanic Sep 24 '23 at 11:17
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$\quad{\cal K}(G)=\alpha_e^{-1}\theta(e!\in!G){\cal K}(G/e)+{\cal K}(G\backslash e)$; $\quad{\cal U}(G)=\theta(e!\in!G){\cal U}(G/e)+\alpha_e{\cal U}(G\backslash e)$; $\quad{\cal F}0(G)=\theta(e!\in!G){\cal F}_0(G/e)+\alpha_e{\cal F}_e(G\backslash e)$; $\quad\det({\cal L}^{\prime}(G))=\text{affine in }\alpha{12}^{-1}=\alpha_{12}^{-1}\det({\cal L}^{\prime}(G/e_{12}))+\det({\cal L}^{\prime}(G\backslash e_{12}))$; The first term adds row 1 to row 2 and column 1 to column 2. For ${\cal F}0(G)$ induction wrt. $\alpha{12}$ in $\det({\cal L}^{\prime n-1}_{n-2}(G))$. – Qmechanic Sep 24 '23 at 11:17
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Assume $G$ connected $|\pi_0(G)|=1$ while $\gamma\subseteq G$ not nec. conn. and might have loops. Then $\quad {\cal K}{n,0}(G)=\sum{n',n^"\in\mathbb{N}0}{\cal K}{n',0}(\gamma){\cal K}{n^",n'+n^"-n}(G/\gamma)$; $\quad\bar{\cal K}{n,\ell}(G):=\sum_{m\in\mathbb{N}0}{\cal K}{n-m,\ell-m}(G)$ $=\sum_{n',n^",\ell',\ell^"\in\mathbb{N}0}^{n-\ell=n'-\ell'+n^"-\ell^"}{\cal K}{n',\ell'}(\gamma){\cal K}{n^",\ell^"}(G/\gamma)$ $=\sum{m\in\mathbb{N}0}\bar{\cal K}{m,\ell}(\gamma)\bar{\cal K}{n,m}(G/\gamma)$ where $\ell'\leq\ell, n^"\leq n$; UV:$\alpha{\gamma}\ll1$. IR:$\alpha_{G/\gamma}\gg1$. – Qmechanic Sep 24 '23 at 14:41
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${\cal K}{00}(G)-{\cal K}{00}(\gamma){\cal K}{00}(G/\gamma)={\cal O}\left((\alpha{G/\gamma}^{-1})^{V(G/\gamma)}\right)$ only contain less powers of $\alpha_{\gamma}^{-1}$ /more powers of $\alpha_{G/\gamma}^{-1}$. $\quad{\cal U}{00}(G)-{\cal U}{00}(\gamma){\cal U}{00}(G/\gamma)={\cal O}\left(\alpha{\gamma}^{L(\gamma)+1}\right)$; $\quad{\cal F}{00}(G)-{\cal U}{00}(\gamma){\cal F}{00}(G/\gamma)={\cal F}{00}(\gamma){\cal U}{00}(G/\gamma)+{\cal U}{10}(\gamma){\cal F}{01}(G/\gamma)+{\cal O}\left(\alpha{\gamma}^{L(\gamma)+2}\right)={\cal O}\left(\alpha_{\gamma}^{L(\gamma)+1}\right)$; – Qmechanic Sep 24 '23 at 15:13
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Proof: $\quad|\pi_0(G/\gamma)|=1$; $\quad V(G)=V(\gamma)-|\pi_0(\gamma)|+V(G/\gamma)$; $\quad E(G)=E(\gamma)+E(G/\gamma)$; $\quad L(G)=L(\gamma)+L(G/\gamma)$; Let $T\subseteq G$ be a spanning $(n+1,\ell)$-subgraph with $V(T)=V(G)$; $|\pi_0(T)|=n+1$; $L(T)=\ell$; $E(T)=V(G)+\ell-n-1$. Then $\tau=T|_{\gamma}$ is a spanning $(n'+|\pi_0(\gamma)|,\ell')$-subgraph in $\gamma$ [since $V(\tau)=V(\gamma)$ all vertices in $\gamma$ belongs to $\tau$] with $n'=|\pi_0(\tau)|-|\pi_0(\gamma)|\geq 0$; $0\leq L(\tau)=\ell'\leq\ell$; $E(\tau)=V(\gamma)+\ell'-n'-|\pi_0(\gamma)|$. – Qmechanic Oct 02 '23 at 08:31
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And then $t=T/\gamma$ is spanning $(n^"+1,\ell^")$-subgraph $G/\gamma$ with $V(t)=V(G/\gamma)$; $n^"+1=|\pi_0(t)|\leq|\pi_0(T)|=n+1$; $0\leq L(t)=\ell^"$; $E(\tau)=V(G/\gamma)+\ell^"-n^"-1$. $\quad V(T)=V(\tau)-|\pi_0(\gamma)|+V(t)$; $\quad E(T)=E(\tau)+E(t)$; $\quad0\leq\ell^"-\ell+\ell'=L(t)-L(T)+L(\tau)=|\pi_0(\tau)|-|\pi_0(\gamma)|-|\pi_0(T)|+|\pi_0(t)|=n^"-n+n'$; $\Box$ – Qmechanic Oct 02 '23 at 08:32
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$G_m$=subgraph of all massive propagators in $G$ (with nec. vertices). $G_p$=subgraph of all external momentum vertices in $G$. Let $\gamma\subseteq G$ be a subgraph. $\quad\gamma$ $m$-cover of $G\Leftrightarrow\gamma\supseteq G_m$. $\quad\gamma$ $p$-cover of $G\Leftrightarrow\gamma\supseteq G_p$. $\quad\gamma$ $p$-connected$\Leftrightarrow\gamma_p\subseteq$ conn. comp. of $\gamma$. $\quad G\backslash\gamma$ IR subgraph of $G\Leftrightarrow\gamma$ $mp$-cover of $G$ and [each conn. comp. of] $\gamma/(G_m\cup G_p)$ is 1PI. (Hm. Speculate whether to replace $G\backslash\gamma$ with $G/\gamma$?) – Qmechanic Oct 03 '23 at 11:48
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$\quad G\backslash\gamma$ IR-irreducible subgraph of $G\Leftrightarrow G\backslash\gamma$ IR and $G/\gamma$ 1VI. $\quad\gamma$ motic$\Leftrightarrow\forall\mu$ proper subgraph that is $p$-connected and $mp$-cover of $\gamma: L(\mu)<L(\gamma)$. – Qmechanic Oct 04 '23 at 11:49
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- Relation to resistor networks https://mathoverflow.net/q/364948 $\alpha_e=$resistance in an edge $e$. $\alpha^{-1}e=$conductance in an edge $e$. Ohm's law $\quad-j_a={\cal A}^{-1}{ab}{\cal E}{b,v} U_v$; current conv $\quad {\cal E}{u,a}j_a=\delta_{u,v}-\delta_{u,w}$; $\quad e_v={\cal L}^{\prime}U^{\prime}$; Resistance $\quad R_{vw}=\frac{\det({\cal L}^{v,w}{v,w})}{\det({\cal L}^{w}{w})}$ from Cramer's rule;
– Qmechanic Oct 04 '23 at 11:49