CALT-2015-024

Non-renormalization Theorems without Supersymmetry

Clifford Cheung and Chia-Hsien Shen Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena, CA 91125∗ (Dated: May 11, 2015) We derive a new class of one-loop non-renormalization theorems that strongly constrain the running of higher dimension operators in a general four-dimensional quantum field theory. Our logic follows from unitarity: cuts of one-loop amplitudes are products of tree amplitudes, so if the latter vanish then so too will the associated divergences. Finiteness is then ensured by simple selection rules that zero out tree amplitudes for certain helicity configurations. For each operator we define holomorphic and anti-holomorphic weights, (w, w) = (n − h, n + h), where n and h are the number and sum over helicities of the particles created by that operator. We argue that an operator Oi can only be renormalized by an operator Oj if wi ≥ wj and wi ≥ wj , absent non-holomorphic Yukawa couplings. These results explain and generalize the surprising cancellations discovered in the renormalization of dimension six operators in the standard model. Since our claims rely on unitarity and helicity rather than an explicit symmetry, they apply quite generally.

INTRODUCTION where n(A) and h(A) are the number and sum over helic- ities of the external states. Since A is physical, its weight Technical naturalness dictates that all operators not is field reparameterization and gauge independent. The forbidden by symmetry are compulsory—and thus gen- weights of an operator O are then invariantly defined by erated by renormalization. Softened ultraviolet diver- minimizing over all amplitudes involving that operator: gences are in turn a telltale sign of underlying symmetry. w(O) = min{w(A)} and w(O) = min{w(A)}. In prac- This is famously true in supersymmetry, where holomor- tice, operator weights are fixed by the leading non-zero phy enforces powerful non-renormalization theorems. contact amplitude2 built from an insertion of O, In this letter we derive a new class of non- w(O) = n(O) − h(O), w(O) = n(O) + h(O), (4) renormalization theorems for non-supersymmetric the- ories. Our results apply to the one-loop running of where n(O) is the number of particles created by O and the leading irrelevant deformations of a four-dimensional h(O) is their total helicity. For field operators we find: quantum field theory of marginal interactions, ¯ ¯ O Fαβ ψα φ ψα˙ Fα˙ β˙ X ∆L = ciOi, (1) h +1 +1/2 0 −1/2 −1 i (w, w) (0, 2) (1/2, 3/2) (1, 1) (3/2, 1/2) (2, 0) where Oi are higher dimension operators. At leading where all Lorentz covariance is expressed in terms of order in ci, renormalization induces operator mixing via four-dimensional spinor indices, so e.g. the gauge field 2 dci X ¯ (4π) = γ c , (2) strength is F ˙ = Fαβ¯ ˙ + F ˙ αβ. The weights of d log µ ij j ααβ˙ β α˙ β α˙ β j all dimension five and six operators are shown in Fig.1. where by dimensional analysis the anomalous dimension As we will prove, an operator Oi can only be renormal- ized by an operator Oj at one-loop if the corresponding matrix γij is a function of marginal couplings alone. The logic of our approach is simple and makes no ref- weights (wi, wi) and (wj, wj) satisfy the inequalities arXiv:1505.01844v1 [hep-ph] 7 May 2015 erence to symmetry. Renormalization is induced by log wi ≥ wj and wi ≥ wj, (5) divergent amplitudes, which by unitarity have kinematic and all Yukawa couplings are of a “holomorphic” form cuts equal to products of on-shell tree amplitudes [1]. If consistent with a superpotential. This implies a new any of these tree amplitudes vanish, then so too will the class of non-renormalization theorems, divergence. Crucially, many tree amplitudes are zero due to helicity selection rules, which e.g. forbid the all minus γij = 0 if wi < wj or wi < wj, (6) helicity gluon amplitude in Yang-Mills theory. which dictate mostly zero entries in the anomalous di- For our analysis, we define the holomorphic and anti- mension matrix. The resulting non-renormalization the- 1 holomorphic weight of an on-shell amplitude A by orems for all dimension five and six operators are shown w(A) = n(A) − h(A), w(A) = n(A) + h(A), (3) in Tab.I and Tab.II.

1 Holomorphic weight is a generalization of k-charge in super 2 By definition, all covariant derivatives D are treated as partial Yang-Mills theory, where the NkMHV amplitude has w = k + 4. derivatives ∂ when computing the leading contact amplitude. 2

dimension 6 In a renormalizable theory, [g] = 0 or 1, so we obtain

dimension 5 F 22 w3, w3 ≥ 2, (9) 6 F 3 F 2 23 6 4 as a lower bound for the three-point amplitude. 2 F 2 2 5 5 2 Next, consider the four-point tree amplitude. As we F ¯ 2D 4 ¯2 2 ¯23 will see, w4, w4 ≥ 4 for the vast majority of amplitudes. 4 2 D The reason is w4 < 4 or w4 < 4 requires non-zero total w 3 ¯22 w F¯22 helicity, which is usually forbidden by helicity selection 2 F¯ ¯2 rules. To show this, we run through all possible candi- ¯4 ¯2 1 F date amplitudes with w4 < 4. Analogous arguments of F¯ ¯2 course apply for w4 < 4. 0 F¯3 1 3 5 Most four-point tree amplitudes with w4 = 1 or 3 w vanish because they have no Feynman diagrams, so 0 2 4 6 w 0 = A(F +F +F ±φ) = A(F +F +ψ±ψ±) Figure 1. Weight lattice for dimension five and six operators, = A(F +F −ψ+ψ+) = A(F +ψ+ψ−φ) suppressing flavor and Lorentz structures, e.g. on which fields + + + − derivatives act. The non-renormalization theorems in Eq. (5) = A(ψ ψ ψ ψ ). let operators to mix into operators of equal or greater weight. Pictorially, this forbids transitions down or to the left. Furthermore, most amplitudes with w4 = 0 or 2 vanish due to helicity selection rules, so 0 = A(F +F +F +F ±) = A(F +F +ψ+ψ−) Since our analysis hinges on unitarity and helicity, the + + + + + resulting non-renormalization theorems are general and = A(F F φ φ) = A(F ψ ψ φ). not derived from an explicit symmetry of the off-shell La- While these amplitudes have Feynman diagrams, they grangian. Moreover, our findings explain the ubiquitous vanish on-shell for their chosen helicities. This leaves a and surprising cancellations [2] observed in the one-loop handful of amplitudes that can in principle be non-zero, renormalization of dimension six operators in the stan- + + + + + + + dard model [3–6]. Lacking an explanation from power 0 6= A(ψ ψ ψ ψ ),A(F φ φ φ),A(ψ ψ φ φ), counting or spurions, the authors of [2] conjectured a hid- for which w4 = 2, 3, 3, respectively. These “exceptional den “holomorphy” enforcing non-renormalization theo- amplitudes” are the only four-point tree amplitudes with rems among holomorphic and anti-holomorphic opera- w4 < 4 that are not identically zero. tors. We show here that this classification simply corre- Since the exceptional amplitudes require external or sponds to w < 4 and w < 4, so the observed cancellations internal scalars, they never arise in theories of only gauge follow directly from Eq. (6), as shown in Tab.II. bosons and fermions, e.g. QCD. The second and third amplitudes require super-renormalizable cubic scalar in- teractions, which we do not consider here. Meanwhile, WEIGHING TREE AMPLITUDES the first amplitude arises from Yukawa couplings of non- holomorphic form, which is to say a combination of cou- We now compute the holomorphic and anti- plings of the form φψ2 together with φψ¯ 2. In a super- holomorphic weights (wn, wn) for a general n-point on- symmetric theory, such couplings would violate holo- shell tree amplitude in a renormalizable theory of mass- morphy of the superpotential. In the standard model, less particles. We start with lower-point amplitudes and Higgs doublet exchange generates an exceptional ampli- then apply induction to extend to higher-point. tude proportional to the product up-type and down-type Working in spinor helicity variables, we consider the Yukawa couplings. This diagram will be important later three-point amplitude with coupling constant g, when we discuss renormalization in the standard model.  r3 r1 r2 P h12i h23i h31i , hi ≤ 0 In summary, we find A(1h1 2h2 3h3 ) = g i (7) r3 r1 r2 P [12] [23] [31] , i hi ≥ 0 w4, w4 ≥ 4, (10) which corresponds to MHV and MHV kinematics, |1] ∝ as a lower bound for the four-point amplitude, modulo |2] ∝ |3] and |1i ∝ |2i ∝ |3i. Lorentz invariance fixes the exceptional amplitudes. P P Finally, consider a general higher-point tree ampli- the exponents to be ri = −ri = 2hi − h and i ri = P tude, A , which on a factorization channel degenerates i ri = 1 − [g] by dimensional analysis [7]. According i to Eq. (7), the weights of the three-point amplitude are into a product of amplitudes, Aj and Ak, where

 P i X h −h (4 − [g], 2 + [g]), i hi ≤ 0 fact[Ai] = Aj(` )Ak(−` ), (11) (w3, w3) = P (8) `2 (2 + [g], 4 − [g]), i hi ≥ 0 h 3

Oi is radiatively generated by Oj. In practice, γij is factorize loop extracted from the one-loop amplitude Ai involving an insertion of Oj that has precisely the same external wi = wj + wk 2 states as the tree amplitude Ai involving an insertion Ai Aj Ak loop of Oi. Any ultraviolet divergence in Ai must be ab- sorbed by the counterterm Ai. The Passarino-Veltman (PV) reduction [8] of the one- cut loop loop amplitude Ai is w = w + w 4 i j k loop X X X loop A A Ai = d4I4 + d3I3 + d2I2 + rational, Ai j k box triangle bubble

Figure 2. Diagrams of tree factorization and one-loop unitar- summing over topologies of scalar box, triangle, and bub- ity, with the weight selection rules from Eqs. (12) and (18). ble integrals, I4, I3, and I2. Tadpole integrals vanish in massless limit considered here. The integral coefficients depicted in Fig.2. If the total numbers and helicities of d4, d3, and d2 are rational functions of external kine- A , A , and A , are (n , h ), (n , h ), and (n , h ), then matic data. Ultraviolet log divergences arise from the i j k i i j j k k scalar bubble integrals in the PV reduction, where in ni = nj + nk − 2 and hi = hj + hk since each side of the 2 factorization channel has an equal and opposite helicity. dimensional regularization, I2 → 1/(4π) . Separating ultraviolet divergent and finite terms, we find Thus, the corresponding weights, (wi, wi), (wj, wj), and

(wk, wk), satisfy the following tree selection rule, loop 1 X Ai = 2 d2 + finite, (14) w = w + w − 2 (4π)  tree rule: i j k (12) bubble wi = wj + wk − 2 which implies a counterterm tree amplitude, We have already shown that w3, w3 ≥ 2 and w4, w4 ≥ 4 1 X modulo the exceptional diagrams. Since all five-point A = − d , (15) i (4π)2 2 amplitudes factorize into three and four-point ampli- bubble tudes, Eq. (12) implies that w5, w5 ≥ 4. Induction to loop higher-point then yields the main result of this section, such that the sum, Ai + Ai, is finite. With generalized unitarity [1], integral coefficients can  2, n = 3 wn, wn ≥ (13) be constructed by relating kinematic singularities of the 4, n > 3 one-loop amplitude to products of tree amplitudes. In which, modulo exceptional amplitudes, is a lower bound particular, the two-particle cut in a particular channel is on the weights of n-point tree amplitudes in a theory of loop X h1 h2 −h1 −h2 massless particles with marginal interactions. Note that cut[Ai ] = Aj(`1 , `2 )Ak(−`1 , −`2 ), (16) h1,h2 even when exceptional amplitudes exist, wn, wn ≥ 2. An important consequence of Eq. (12) is that attach- where `1, `2 and h1, h2 are the momenta and helicities of ing renormalizable interactions to any amplitude Aj, the cut lines and Aj and Ak are on-shell tree amplitudes even involving irrelevant interactions, can only produce corresponding to the cut channel, as depicted in Fig.2. an amplitude Ai of greater or equal weight. To see Applying this same cut to the PV reduction, we find why, note that Ai factorizes into Aj and an amplitude A composed of only renormalizable interactions, where loop k cut[Ai ] = d2 + terms that depend on `1, `2, (17) wk, wk ≥ 2 by Eq. (13). Eq. (12) then implies that wi ≥ wj and wi ≥ wj, so the minimum weight ampli- where the `1, `2 dependent terms come from two-particle tude involving a higher dimension operator is the contact cuts of triangle and box integrals. As is well-known, the amplitude built from a single insertion of that operator. divergence of the one-loop amplitude is related to the two-particle cut [9–11]. However, a kinematic singular- ity is present only if Aj and Ak are four-point amplitudes WEIGHING ONE-LOOP AMPLITUDES or higher, corresponding to “massive” bubble integrals. When Aj or Ak are three-point amplitudes, the associ- We now calculate the weights of one-loop amplitudes ated “massless” bubble integrals are scaleless and vanish using generalized unitarity and the tree-level results of in dimensional regularization. For now we ignore these the previous section. Leading order renormalization subtle contributions but revisit them in the next section. of higher dimension operators is characterized by the Combining Eq. (15) with Eqs. (16) and (17), we find anomalous dimension matrix γij, which encodes how that the total numbers and helicities (ni, hi), (nj, hj), 4

F 2φ F ψ2 ψ2φ2 F¯ψ¯2 F¯2φ ψ¯2φ2 φ5 inclusive sum over final states corresponding to tree-level (w, w¯) (1, 5) (1, 5) (3, 5) (5, 1) (5, 1) (5, 3) (5, 5) real emission of an unresolved soft or collinear particle. 2 F φ (1, 5) Inverting the logic, if real emission is actually infrared F ψ2 (1, 5) finite, then there can be no virtual infrared divergence ψ2φ2 (3, 5) and thus no ultraviolet divergence. As we will show, F¯2φ (5, 1) F¯ψ¯2 (5, 1) this is true of the discarded contributions from massless ψ¯2φ2 (5, 3) bubbles which could have a priori violated Eq. (5). 5 φ (5, 5) loop To diagnose potential infrared divergences in Ai , we real Table I. Anomalous dimension matrix for dimension five op- analyze the associated amplitude for real emission, Ai0 . erators in a general quantum field theory. The shaded entries In the infrared regime, the singular part of this ampli- vanish by our non-renormalization theorems. real tude factorizes: Ai0 → AiSi→i0 + AjSj→i0 , where Ai and Aj are tree amplitudes built around insertions of Oi and O , and S 0 and S 0 are soft-collinear functions (n , h ) of A , A and A satisfy n = n + n − 4 and j i→i j→i k k i j k i j k describing the emission of an unresolved particle. Since h = h + h . This implies the one-loop selection rule, i j k infrared divergences are a long distance effect, we only w = w + w − 4 consider the soft-collinear functions for emissions gener- one-loop rule: i j k (18) wi = wj + wk − 4 ated√ by marginal interactions. They diverge as 1/ω and 1/ 1 − cos θ in the soft and collinear limits, respectively, where (wi, wi), (wj, wj), and (wk, wk) are the weights where ω and θ are the energy and splitting angle char- of Ai, Aj, and Ak, respectively. For the entry γij of acterizing the emitted particle. Since the phase-space the anomalous dimension matrix, we identify Ai and Aj measure is dω ω d cos θ, infrared divergences require with tree amplitudes built around insertions of O and i that Si→i0 and´ Sj→´i0 are both soft and/or collinear. Oj, and Ak with a tree amplitude of the renormaliz- able theory. As noted earlier, the amplitudes on both For soft emission, the hard process is unchanged [15]. sides of the cut must be four-point or higher to have Since AiSi→i0 and AjSj→i0 contribute to the same pro- a non-trivial unitarity cut, which implies from Eq. (13) cess, this implies that Ai and Aj have the same external that wk, wk ≥ 4 if there are no exceptional amplitudes. states and thus equal weight, wi = wj. While mass- Plugging back into Eq. (18) implies that wi ≥ wj and less bubbles do contribute infrared and ultraviolet diver- wi ≥ wj, which is the non-renormalization theorem in gences not previously accounted for, this is perfectly con- Eq. (5). If exceptional amplitudes are present, say from sistent with the non-renormalization theorem in Eq. (5), non-holomorphic Yukawas, then wk, wk = 2 and Eq. (5) which allows for operator mixing when wi = wj. Vi- is violated, albeit by exactly two units in weight. olation of Eq. (5) requires infrared divergences when The weight lattice for all dimension five and six oper- wi < wj, but this only happens for soft emission that ators in a general quantum field theory is presented in flips the helicity of a hard particle, which is subleading Fig.1. We use the operator basis of [12] in which redun- in the soft limit and thus finite upon dω integration. dant operators, e.g. those involving φ, are eliminated ´ by equations of motion. Our non-renormalization2 the- Similarly, collinear emission is divergent for wi = wj orems imply that operators can only renormalize other but finite for wi < wj. Since AiSi→i0 and AjSj→i0 operators of equal or greater weight, which in Fig.1 for- have the same external states and weight, restricting to bids transitions that move down or to the left. The form wi < wj means that w(Si→i0 ) > w(Sj→i0 ). Eq. (8) then of the anomalous dimension matrix for all dimension five implies that Si→i0 and Sj→i0 are collinear splitting func- and six operators is shown in Tab.I and Tab.II. tions generated by on-shell MHV and MHV amplitudes. ∗ As a result, the interference term Sj→i0 Si→i0 carries net little group weight with respect to the mother particle INFRARED DIVERGENCES initiating the collinear emission. Rotations of angle φ around the axis of the mother particle act as a little ∗ We now return to the issue of massless bubble inte- group transformation on Sj→i0 Si→i0 , yielding a net phase grals. While these contributions formally vanish in di- e2iφ in the differential cross-section. Integrating over this mensional regularization, this is potentially misleading 2π 2iφ angle yields 0 dφ e = 0, so the collinear singularity because ultraviolet and infrared divergences enter with vanishes upon´ phase-space integration. opposite sign 1/ poles. Thus, an ultraviolet divergence may actually be present if there happens to be an equal In summary, for wi < wj we have found that real emis- and opposite virtual infrared divergence [9–11]. Cru- sion is infrared finite, so there are no ultraviolet diver- cially, the Kinoshita-Lee-Nauenberg theorem [14] says gences from massless bubbles. The non-renormalization that all virtual infrared divergences are canceled by an theorems in Eq. (5) apply despite infrared subtleties. 5

F 3 F 2φ2 F ψ2φ ψ4 ψ2φ3 F¯3 F¯2φ2 F¯ψ¯2φ ψ¯4 ψ¯2φ3 ψ¯2ψ2 ψψφ¯ 2D φ4D2 φ6 (w, w¯) (0, 6) (2, 6) (2, 6) (2, 6) (4, 6) (6, 0) (6, 2) (6, 2) (6, 2) (6, 4) (4, 4) (4, 4) (4, 4) (6, 6) 3 F (0, 6) × × × × × × × × × × 2 2 F φ (2, 6) × × × × × × 2 F ψ φ (2, 6) × × × 4 ψ (2, 6) × × × × × × × × y2 × × 2 3 ψ φ (4, 6) ×∗ y2 × 3 F¯ (6, 0) × × × × × × × × × × 2 2 F¯ φ (6, 2) × × × × × × 2 F¯ψ¯ φ (6, 2) × × × 4 ψ¯ (6, 2) × × × × × × × × y¯2 × × 2 3 ψ¯ φ (6, 4) y¯2 ×∗ × 2 2 ψ¯ ψ (4, 4) × y¯2 × × y2 × × × 2 ψψφ¯ D (4, 4) × 4 2 φ D (4, 4) × × × × 6 φ (6, 6) ×∗ × × ×∗ × × ×

Table II. Anomalous dimension matrix for dimension six operators in a general quantum field theory. The shaded entries vanish by our non-renormalization theorems, in full agreement with [2]. Here y2 and y¯2 label entries that are non-zero due to non-holomorphic Yukawa couplings, × labels entries that vanish because there are no diagrams [13], and ×∗ labels entries that vanish by a combination of counterterm analysis and our non-renormalization theorems.

APPLICATION TO THE STANDARD MODEL of tree-level QCD [18], and so merits further study.

Since our results rely on unitarity and helicity, they apply to any four-dimensional quantum field theory of CONCLUSIONS massless particles, including the standard model and its extension to higher dimension operators. Incidentally, We have derived a new class of one-loop non- there has been much progress in this direction in recent renormalization theorems for higher dimension opera- years [2–6]. A tour de force calculation of the full one- tors in a general four-dimensional quantum field theory. loop anomalous dimension matrix of dimension six oper- Since our arguments make no reference to symmetry— ators [4] unearthed a string of miraculous cancellations only unitarity and helicity—they are broadly applicable, not enforced by an obvious symmetry and visible only and explain the peculiar cancellations observed in the after the meticulous application of equations of motion renormalization of dimension six operators in the stan- [2]. Lacking a manifest symmetry of the Lagrangian, dard model. Let us briefly discuss future directions. the authors of [2] conjectured an underlying “holomor- First and foremost is the matter of higher loop or- phy” of the standard model effective theory that ensures ders. As is well-known, helicity selection rules—e.g. the closure of certain operators under renormalization. vanishing of the all minus amplitude in Yang-Mills—are The cancellations in [2] are a direct consequence of violated by finite one-loop corrections [19]. While this the non-renormalization theorems in Eq. (5) and Eq. (6), strongly suggests that Eq. (5) should fail at two-loop or- based on a classification of holomorphic (w < 4), anti- der, this important question deserves close examination. holomorphic (w < 4), and non-holomorphic operators Another natural direction is higher dimensions, where (w, w ≥ 4), and violated only by exceptional amplitudes helicity is naturally extended [20] and dimensional re- (w, w = 2) generated by non-holomorphic Yukawas. The duction offers a bridge to massive theories. Finally, there shaded entries in Tab.II denote zeroes enforced by our is the question of finding concrete linkage between our non-renormalization theorems. Entries marked with × results and more conventional symmetry arguments like vanish trivially because there are no associated Feynman those of [16]. Indeed, our definition of weight is reminis- diagrams, while the few entries marked with ×∗ vanish cent of both R-symmetry and twist, which are known to because the expected divergences in ψ2φ3 and φ6 are ac- relate closely to existing non-renormalization theorems. companied by a counterterm of the form φ4D2 [4] which is forbidden by our non-renormalization theorems. Acknowledgments: We would like to thank Rodrigo Interestingly, the superfield formalism offers an en- Alonso, Zvi Bern, Lance Dixon, Yu-tin Huang, Elizabeth lightening albeit partial explanation of these cancella- Jenkins, David Kosower, and Aneesh Manohar for useful tions [16] as well as analogous effects in chiral perturba- discussions. C.C. and C.-H.S. are supported by a DOE tion theory [17]. These results are clearly connected to Early Career Award under Grant No. DE-SC0010255. our own via the well-known “effective” supersymmetry C.C. is also supported by a Sloan Research Fellowship. 6

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