The IR/UV Connection in the Non-Commutative Gauge Theories

Alec Matusis; Leonard Susskind; Nicolaos Toumbas

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The IR/UV Connection in the Non-Commutative Gauge Theories

Alec Matusis, Leonard Susskind, Nicolaos Toumbas

High Energy Physics - Theory · Submitted 9 Feb 2000 · 9 references

Pith's verdict: The gauge-theory half of the IR/UV story, and it holds up: the 1/theta^2 and 1/theta poles with coefficient (Ns + 2 − 2Nf) are computed explicitly and have survived twenty-five years of rederivation.

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CitationAlec Matusis, Leonard Susskind, Nicolaos Toumbas. "The IR/UV Connection in the Non-Commutative Gauge Theories." arXiv:hep-th/0002075 (2000). https://pith.science/paper/Q2R72BSA

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Short version: this is the gauge-theory companion to the Minwalla–Van Raamsdonk–Seiberg phi^4 paper, and it is the one that made the IR/UV connection a statement about gauge theory rather than about scalars. The 1/theta^2 pole in the photon self-energy (Eq. 3.10) and the 1/theta pole in the three-point vertex (Eq. 4.14), both with coefficient (Ns + 2 − 2Nf), are what everyone since has quoted. The integrals behind them are written out and checkable: the fermion trace reduction to Eq. 3.6, the tensor reduction in Eqs. 4.10–4.13, and the observation that the (p-tilde g) structure cancels identically. Nothing is fitted, nothing is assumed. Credit is given to Hayakawa [8] for overlapping work, which is the right thing to do and they do it plainly. The physics payoff is also real: the two transverse polarizations of the photon get different dispersion relations, Eq. 3.8 versus 3.9, and that splitting is the cleanest statement of what noncommutativity does to the low-energy spectrum. The mass-independence argument for softly broken SUSY is a nice touch and correct — the effect comes from high loop momenta, so a mass term only touches the logs. On the soft spots: Section 5 checks gauge invariance of the anomalous S-matrix in two kinematic corners of one process, gauge boson plus fermions into gauge boson plus fermions, and explicitly defers the scalar case to [9], listed as in preparation. That deferral matters more than it looks, because Ns is exactly the sector whose consistency is unchecked and Ns sits in the headline coefficient. My read is that the same tensor identity that kills the p-tilde-g term in Eq. 4.13 forces the scalar vertex to combine the same way, so I would be surprised if it failed — but surprised is not proved, and the authors say so themselves. The N=4 statement in the abstract is hedged as 'seem to be absent,' which is the correct hedge; it is an expectation, not a derivation, and Section 6 treats it that way. Neither gap touches the pole computation that carries the paper. This is for anyone working on noncommutative field theory, stringy or otherwise, and for anyone who wants the canonical reference for why theta to zero is singular. Send it out.

Referee report

Significance

This is a foundational paper for the IR/UV mixing phenomenon in noncommutative field theory. The one-loop integrals are explicit and self-contained: the fermion trace is reduced in Eqs. (3.1)–(3.4), the Fourier transform of 1/l⁴ is stated and used, and the tensor reduction of the three-point integral is carried through in Eqs. (4.9)–(4.13). The structural result — that the leading poles are controlled by the boson–fermion mismatch N_s + 2 − 2N_f, and that the subleading logs track the β function — is a genuinely predictive statement, and it is checkable: for N=4 SYM the authors' own counting (N_s = 6, N_f = 4) gives 6 + 2 − 8 = 0, consistent with the claimed cancellation. The observation that the anomalous terms are local in the commutative directions and nonlocal only in the noncommutative plane is a useful and durable characterization. The paper is also honest about the limits of its gauge-invariance argument, which is a point in its favour rather than against it.

Major objections (2)

[§5 and §6] The gauge-invariance check of the anomalous terms is carried out for a single process (gauge boson plus fermions into gauge boson plus fermions) and only in two kinematic corners, small p̃ and small l̃. The scalar case is explicitly deferred: 'The case involving scattering of scalars is more subtle because the two scalar – gauge boson vertex also contains linear divergences in theta. We defer this case for an upcoming paper [9].' Reference [9] is listed as 'in preparation', so this part of the consistency argument is not available to the reader. This matters because N_s appears in the headline coefficient of Eqs. (3.10) and (4.14), and the scalar sector is precisely the one whose gauge consistency is left unchecked. The authors should either extend the §5 argument to include the scalar–gauge vertex, or state plainly in §5 and in the Conclusions that gauge invariance of the anomalous amplitude has been established only for the gauge and fermion sectors, with the N_s entry in Eqs. (3.10) and (4.14) resting on the vertex computation alone. As written, the sentence in §6 that 'the structure of these new poles is consistent with gauge invariance' reads as covering the full coefficient, which the §5 analysis does not establish.
[§3, Eqs. (3.5)–(3.6)] The step from Eq. (3.5) to Eq. (3.6) is the crux of transversality and is compressed into the single sentence 'the term g_{μν}/p̃² cancels in this expression'. Because p̃ is restricted to the (1,2) plane, differentiating log|p̃| with respect to the four-dimensional p̃ does not automatically produce a clean g_{μν} piece, and the reader cannot verify the stated cancellation without redoing the regulated integral. Since the transversality of Eq. (3.6) (via p·p̃ = 0) is what makes the result compatible with gauge invariance and what underlies the whole of §5, the algebra should be displayed explicitly, including how the regulator-dependent pieces are handled. The same request applies to the analogous cancellation noted after Eq. (4.13), where the second tensor structure is said to cancel 'completely'.

Minor objections (5)

[§6 and Abstract] The paper should state explicitly that the 1/p̃² and 1/p̃ poles cannot be absorbed by local counterterms, because they are non-analytic in θ and therefore not of the form of a local term in the Lagrangian. This is implicit in the 'anomaly' language and in the discussion of the singular θ → 0 limit, but making it explicit would sharpen the central claim and preempt the natural question of whether the effect is a scheme artifact.
[§3, Eq. (3.1) and the definition of N_f] The convention 'N_f is the number of Majorana fermions, each of which counts as two fermion species' is confusing when read against Eq. (3.6), which carries an explicit factor of (2N_f), and against Eq. (3.10), where the same combination appears inside (N_s + 2 − 2N_f). A single sentence fixing the counting convention once, with the N=4 values N_s = 6 and N_f = 4 quoted as a check, would remove the ambiguity.
[§4, Eq. (4.1) and Figures 4–5] The figure references ('Fig. 5, 1st graph on line (1)', 'Second graph and its two permutations on line (1) in Fig. 5') are difficult to follow, and the symmetry factors of 1/2 for the quartic graphs and the factor of 2 for the ghost and fermion graphs are asserted rather than derived. A short enumeration of the diagrams with their symmetry factors would make the coefficient (N_s + 2 − 2N_f) in Eq. (4.8) independently checkable.
[§5, Eq. (5.9)] The cancellation of the one-loop amplitude against the tree-level graph uses the corrected dispersion relation p² = 8αg²/p̃² taken from the paper's own Eq. (3.10). This is the correct procedure in perturbation theory, but it should be said explicitly that the §5 check is a self-consistency test of the pole structure rather than an independent determination of the coefficient.
[Throughout] Typographical and presentational items: 'thoery' in the caption of Figure 1; 'We refer the reader in [6][7]' should read 'to [6][7]'; 't'Hooft' should be '’t Hooft'; 'Pauli Villars' should be 'Pauli–Villars'; 'depend only' appears duplicated in the sentence beginning 'The matrix elements in eqs(3.10) and (4.14)'; spacing in 'Eq(2.5)' and 'eq(3.1)' is inconsistent. The Feynman rules in the Appendix should be typeset as displayed equations rather than as embedded graphics, since several of the signs and index contractions are hard to read in the current form. Reference [9], listed as 'in preparation', should be updated at proof if it has appeared.

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The paper

The paper claims

In noncommutative gauge theory, high-momentum loops leave poles at low momentum, so the limit of vanishing noncommutativity is singular.

The reading

Noncommutative field theories are built with a star product whose parameter $theta$ has dimensions of length squared; classically, momenta well below $theta^{-1/2}$ see an ordinary local theory. This paper asks whether that classical equivalence survives quantization, and answers no for gauge theory at one loop. Computing the photon self-energy and the three-photon vertex in noncommutative $U(1)$ Yang-Mills, the authors find that non-planar diagrams, whose oscillating phases $e^{ip tilde l}$ soften the ultraviolet, leave behind poles in $tilde p = theta p$: a $1/tilde p^2$ term in the two-point function and a $1/tilde p$ term in the three-point function. The coefficients come out proportional to $N_s + 2 - 2N_f$, the excess of bosonic over fermionic loop species, so the poles cancel in supersymmetric theories while the $theta to 0$ limit stays singular otherwise. The authors then check, in a two-fermion two-gauge-boson scattering process, that the anomalous terms are consistent with the theory's Ward identities, and they acknowledge that the corresponding scalar check is deferred to a companion paper still in preparation.

What carries the argument

The load-bearing mechanism is the phase factor $e^{i p theta l}$ attached to each non-planar diagram, which at large loop momentum $l$ oscillates and renders finite an integral that would be divergent in the commutative theory, but which becomes ineffective as the external noncommutative momentum $tilde p = theta p$ goes to zero. The computation is carried by the Fourier transform identity $int d^4 l / (2pi)^4, l^{-4} e^{i tilde p l} = i alpha (log Lambda - log |tilde p|)$ and its descendants obtained by differentiating with respect to $tilde p$, which convert the regulated short-distance behaviour into explicit $log |tilde p|$ and hence into poles in $tilde p$; the resulting tensor structures $tilde p^mu tilde p^nu / tilde p^4$ and $tilde p^{mu_1} tilde p^{mu_2} tilde p^{mu_3} / tilde p^4$ are automatically transverse in the sense of the Ward identity, which is what makes the anomalous terms compatible with gauge invariance despite breaking Lorentz invariance.

Extended reading notes

Core claim

The paper's central claim is that one-loop non-planar diagrams in noncommutative gauge theory generate infrared singularities that scale as inverse powers of the noncommutativity parameter, contradicting the expectation that a theory whose worst commutative divergences are logarithmic should produce at most logarithmic anomalous $tilde p$ dependence. Concretely, the photon vacuum polarization in noncommutative $U(1)$ Yang-Mills contains a term $iPi^{mu nu}(p) = 8 i g^2 (N_s + 2 - 2N_f) alpha, tilde p^mu tilde p^nu / tilde p^4$, which is transverse (since $p cdot tilde p = 0$) but splits the dispersion relation of the two physical photon polarizations: the mode polarized perpendicular to $tilde P$ keeps $p_0^2 = p_3^2 + P^2$, while the mode polarized along $tilde P$ obeys $p_0^2 = p_3^2 + P^2 + c g^2 / (theta^2 P^2)$. The three-photon vertex carries a weaker but still non-analytic pole, $Gamma^{mu_1 mu_2 mu_3} = -8 alpha g^3 (N_s + 2 - 2N_f) cos(tfrac{1}{2} tilde p_3 p_1) sum_i tilde p_i^{mu_1} tilde p_i^{mu_2} tilde p_i^{mu_3} / tilde p_i^4$ plus terms at most logarithmic. Because the coefficient counts bosons minus fermions, all inverse-power poles disappear in supersymmetric models, and the authors argue that noncommutative super Yang-Mills with 16 real supercharges is free of the anomalous effects altogether, being the one known example that reduces to its commutative counterpart as $theta to 0$.

Load-bearing premise

The claim that the anomalous terms respect gauge invariance rests on a check in Section 5 that covers only two kinematic corners of one process, gauge bosons plus fermions scattering to gauge bosons plus fermions, and the authors themselves acknowledge that the scalar version of this check is deferred to a companion paper listed as in preparation.

Editorial extensions

If this is right

The commutative limit of a noncommutative gauge theory is not smooth: physical quantities such as the photon dispersion relation diverge as $theta to 0$, so noncommutativity cannot be treated as a small perturbation of ordinary Yang-Mills theory.
The two photon polarizations acquire different dispersion relations, a Lorentz-violating splitting that survives because $theta$ itself picks out a preferred plane; the effect is local in the directions orthogonal to that plane and nonlocal within it.
Inverse-power poles cancel between bosons and fermions at one loop, so any supersymmetric noncommutative gauge theory has at worst logarithmic anomalous dependence on $tilde p$, tied to the coefficient of the commutative beta function.
Noncommutative super Yang-Mills with 16 real supercharges appears to be completely free of these anomalies, making it the natural candidate for a theory with a well-behaved $theta to 0$ limit.
A theory with more fermions than bosons gets a negative coefficient, which the authors note destabilizes the low-energy theory, paralleling the behaviour of $phi^4$ with Yukawa-coupled fermions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the inverse-power poles are real, $theta$ is not a continuously deformable parameter of the quantum theory but a phase-labeling one: the noncommutative and commutative theories would sit in different phases, which suggests treating noncommutativity as fundamental rather than as a small correction to be expanded in.
The cancellation condition $N_s + 2 - 2N_f = 0$ looks like a one-loop coefficient of its own, distinct from the ordinary beta function; it would be worth asking whether this same combination controls the subleading logarithmic terms, which would make the pole coefficient a genuine one-loop invariant of the noncommutative theory.
Because the anomalous terms are transverse yet Lorentz-violating, the polarization splitting is the cleanest observable signature: in any setting where $theta$ arises as a background field, the two photon polarizations propagate with measurably different velocities at low momentum.
The authors' gauge-invariance check covers only two kinematic corners of a single scattering process; the natural extension is a full all-kinematics Ward identity check, and separately the scalar version they defer, since scalar loops enter with their own cubic and quartic vertices.

Original abstract

Quantum field theory on non-commutative spaces does not enjoy the usual ultraviolet-infrared decoupling that forms the basis for conventional renormalization. The high momentum contributions to loop integrations can lead to unfamiliar long distance behavior which can potentially undermine naive expectations for the IR behavior of the theory. These "anomalies" involve non-analytic behavior in the noncommutativity parameter Theta making the limit Theta goes to zero singular. In this paper we will analyze such effects in the one loop approximation to gauge theories on non-commutative space. We will see that contrary to expectations poles in Theta do occur and lead to large discrepancies between the expected and actual infrared behavior. We find that poles in Theta are absent in supersymmetric theories. The "anomalies" are generally still present, but only at the logarithmic level. A notable exception is non-commutative super Yang Mills theory with 16 real supercharges in which anomalous effects seem to be absent altogether.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Quantum field theory on non-commutative spaces does not enjoy the usual ultraviolet-infrared decoupling that forms the basis for conventional renormalization. The high momentum contributions to loop integrations can lead to unfamiliar long distance behavior which can potentially undermine naive expectations for the IR behavior of the theory. These "anomalies" involve non-analytic behavior in the noncommutativity parameter Theta making the limit Theta goes to zero singular. In this paper we will analyze such effects in the one loop approximation to gauge theories on non-commutative space. We will see that contrary to expectations poles in Theta do occur and lead to large discrepancies between the expected and actual infrared behavior. We find that poles in Theta are absent in supersymmetric theories. The "anomalies" are generally still present, but only at the logarithmic level. A notable exception is non-commutative super Yang Mills theory with 16 real supercharges in which anomalous effects seem to be absent altogether.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Quantum field theory on non-commutative spaces does not enjoy the usual ultraviolet-infrared decoupling that forms the basis for conventional renormalization.	background 1.00	0.05	0.05	asserted 1.00
s1	The high momentum contributions to loop integrations can lead to unfamiliar long distance behavior which can potentially undermine naive expectations for the IR behavior of the theory.	background 1.00	0.06	0.04	hedged possibility 0.95
s2	These "anomalies" involve non-analytic behavior in the noncommutativity parameter Theta making the limit Theta goes to zero singular.	background 0.96	0.10	0.07	asserted 1.00
s3	In this paper we will analyze such effects in the one loop approximation to gauge theories on non-commutative space.	contribution 0.95	0.55	0.04	asserted 0.81
s4	We will see that contrary to expectations poles in Theta do occur and lead to large discrepancies between the expected and actual infrared behavior.	result 0.98	0.80	0.44	asserted 0.99
s5	We find that poles in Theta are absent in supersymmetric theories.	result 1.00	0.83	0.20	asserted 1.00
s6	The "anomalies" are generally still present, but only at the logarithmic level.	result 0.99	0.77	0.14	asserted 0.97
s7	A notable exception is non-commutative super Yang Mills theory with 16 real supercharges in which anomalous effects seem to be absent altogether.	result 1.00	0.58	0.15	hedged possibility 0.94

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

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full text
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Figures

**Figure 1.** Mass renormalization correction in φ 4 thoery. such effects “anomalies” although we emphasize that they do not signal an inconsistency in the theory but rather a violation of naive expectations. In the case of planar diagrams the effect of non-commutativity is simple. Every diagram gets multiplied by a phase factor that depends only on the momenta of the external lines. Thus the Feynman integrals are exactly as in the commutative theory. In the non-planar case the situation is more interesting. The phase factors now involve products of external momenta p and internal momenta l in

**Figure 2.** Photon self energy: gauge loop corrections.

**Figure 3.** Photon self energy: matter loop corrections.

**Figure 4.** Vertex corrections: gauge sector. 4 Vertex Corrections Next we analyze 1–loop corrections to the three–photon vertex. In Figures 4 and 5, we have drawn all 1PI Feynman diagrams that contribute corrections to the vertex up to 1–loop order in perturbation theory. As before, we shall see that there are unfamiliar long distance effects arising from the region of integration of high loop momenta. Let us consider in detail the scalar graph involving cubic vertices only (

**Figure 5.** Vertex corrections: matter sector. commutative case the oscillating phases spoil the rotational symmetry. They make the integral finite but they induce low momentum poles of the form p˜ µ1 p˜ µ2 p˜ µ3 p˜ 4 . (4.4) Unlike the previous effect which was of order θ −2 , this is a θ −1 effect. To compute the precise coefficient in front of such anomalous terms, it is enough to consider high momentum running in the loop ignoring external momenta in the denominators. Then the integral becomes − 8ig3 cos(p˜3p1 2 ) Z d 4 l (2π) 4 8l µ1 l µ2 l µ3 l 6 sin p˜1l 2 ! sin p˜2l 2 ! sin p˜3l 2 ! .…

**Figure 6.** Tree level scattering diagrams. + + p q l

**Figure 7.** The sum of these diagrams is gauge invariant.

Figures reproduced from arXiv:hep-th/0002075 by the authors.

References

Works cited (9)

[8]

M. Hayakawa, “Perturbative analysis on infrared and ultraviolet aspects of non– commutative QED on R**4,” hep–th/9912167, hep–th/9912094

View on Pith
[9]

Alec Matusis, Leonard Susskind, Nicolaos Toumbas, in preparatio n. 17

View on Pith
[1]

A. Connes, M. R. Douglas, A. Schwarz, “Non–commutative geom etry and matrix theory: compactiﬁcation on tori,” JHEP 9802:003,1998, hep–th/97 11162

View on Pith 1998
[2]

Nathan Seiberg, Edward Witten, “String theory and non–commu tative Geometry,” JHEP 9909:032,1999, hep–th/9908142

View on Pith 1999
[3]

Daniela Bigatti, Leonard Susskind, “Magnetic ﬁelds, branes and n on–commutative geometry,” hep–th/9908056

View on Pith
[4]

Sheikh Jabbari, “Open strings in a B–ﬁeld background as electric d ipoles,” hep– th/9901080

View on Pith
[5]

Shiraz Minwalla, Mark Van Raamsdonk, Nathan Seiberg, “Non–com mutative pertur- bative dynamics,” hep–th/9912072

View on Pith
[6]

T. Krajewski, R. Wulkenhaar, “Perturbative quantum gauge ﬁe lds on a non– commutative torus,” hep–th/9903187

View on Pith
[7]

T. Filk, “Divergences in a ﬁeld theory on quantum space,” Phys.Let t.B376 (1996) 53

View on Pith 1996

Forward citations

Cited by 12 Pith papers

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How the literature uses it

77 arXiv papers cite this in 123 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 93 sentences without a confident label

Compares with it 3 papers

However, in the limit $→0$ transverality is recovered, and [ref] reduces to the well-known expression [ref] [equation] In the ultraviolet, the graphs of fig:1loop_3A_all diverge only logarithmically.

Loop Calculations for the Non-Commutative U(1) Gauge Field Model with Oscillator Term · Vertex corrections at one-loop level · 0.86
These results are in agreement with the literature [ref] .

Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences · Only external $U(1)$ legs · 0.90
This is reminiscent of, but different from, the UV/IR connection in noncommutative field theories [ref] .

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Cites it as support 4 papers

In particular, it is argued [ref] and shown in the light-cone gauge [ref] that the commutative limit ($^2_Θ→^2$) of the noncommutative $=4$ SYM theory is continuous.

A one-loop test for construction of 4D N=4 SYM from 2D SYM via fuzzy sphere geometry · Introduction · 0.97
Thus it is (expected to be) UV finite, and free of pathological UV/IR mixing [ref] .

Emergent Geometry and Gravity from Matrix Models: an Introduction · Introduction · 1.00
Particularly well behaved are theories where the gauge fields couple via a star commutator ([ref]) to fermions, as is the case in our model: Cancelations between fermion and boson loops lead to softened UV/IR coupling [ref] .

Photon-neutrino interaction in theta-exact covariant noncommutative field theory · Introduction · 0.98

Cites it as background 22 papers

For a long time quantum field theories formulated in a Groenewold-Moyal deformed (or $þ$-deformed) space [ref] suffered from new types of divergences arising due to a phenomenon referred to as [ref] .

A New Approach to Non-Commutative U(N) Gauge Fields · Introduction · 0.99
This phenomenon was first evidenced on the Moyal space in the context of scalar field theories [ref] and shortly extended to noncommutative gauge theories [ref] .

On the UV/IR mixing of Lie algebra-type noncommutatitive $\phi^4$-theories · Introduction · 0.88
NCFT on noncommutative Moyal spaces received a lot of attention from the end of the 90's, in particular from the viewpoint of perturbative properties and renormalisability [ref] .

Noncommutative gauge theories on $\mathbb{R}^3_\lambda$: Perturbatively finite models · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [8]

M. Hayakawa, “Perturbative analysis on infrared and ultraviolet aspects of non– commutative QED on R**4,” hep–th/9912167, hep–th/9912094

View on Pith

[2] [9]

Alec Matusis, Leonard Susskind, Nicolaos Toumbas, in preparatio n. 17

View on Pith

[3] [1]

A. Connes, M. R. Douglas, A. Schwarz, “Non–commutative geom etry and matrix theory: compactiﬁcation on tori,” JHEP 9802:003,1998, hep–th/97 11162

View on Pith 1998

[4] [2]

Nathan Seiberg, Edward Witten, “String theory and non–commu tative Geometry,” JHEP 9909:032,1999, hep–th/9908142

View on Pith 1999

[5] [3]

Daniela Bigatti, Leonard Susskind, “Magnetic ﬁelds, branes and n on–commutative geometry,” hep–th/9908056

View on Pith

[6] [4]

Sheikh Jabbari, “Open strings in a B–ﬁeld background as electric d ipoles,” hep– th/9901080

View on Pith

[7] [5]

Shiraz Minwalla, Mark Van Raamsdonk, Nathan Seiberg, “Non–com mutative pertur- bative dynamics,” hep–th/9912072

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