University of Wisconsin - Madison
한글 요약 계속:
1-loop에서 Higgs decay와 process γγ →HH의 결과는 large logarithmic terms으로 인해 기존의 등가 이론(approximation)이 효과적으로 작동하지 않는다는 것을 보여준다. 본 논문에서는 1-loop에서 large logarithms를 포함하는 등가 이론을 테스트하고 실제 계산과 비교한다.
한글 요약 마침:
이번 연구는 중력의 힉스(decay H →γγ와 process γγ →HH)에 대한 등가 이론(equivalence theorem)의 효과성을 검토했다. 본 논문에서 제시된 결과는 1-loop에서 large logarithmic terms으로 인해 기존의 등가 이론(approximation)이 효과적으로 작동하지 않는다는 것을 보여준다.
영어 요약 시작:
The paper discusses the test of an equivalence theorem at one-loop for Higgs decays and processes involving photons, specifically focusing on the Higgs decay to two photons (H →γγ) and the process γγ →HH. The known 1-loop calculations contain large logarithms, which make the equivalence theorem a poor approximation.
영어 요약 계속:
The study reveals that the results of one-loop calculations for these processes are affected by large logarithmic terms, making the standard equivalence theorem less effective than expected. This is demonstrated through explicit calculations and comparisons with actual computations.
영어 요약 마침:
This research explores the effectiveness of an equivalence theorem at one-loop for Higgs decays and photon-related processes, specifically highlighting its limitations due to large logarithmic terms in one-loop calculations.
University of Wisconsin - Madison
arXiv:hep-ph/9207275v1 30 Jul 1992University of Wisconsin - MadisonMAD/PH/712July 1992Test of an Equivalence Theorem at One-LoopM. S. BergerPhysics Department, University of Wisconsin, Madison, WI 53706, USAABSTRACTWe show that the equivalence theorem approximating one-loop gauge sectordiagrams by including only Goldstone bosons in the loop gives a remarkablypoor approximation to the amplitude for the decay H →γγ and for the processγγ →HH.
At one loop, large logarithms can arise that evade power countingarguments.
The standard equivalence theorem [1] has become a popular method for approximat-ing difficult calculations. Amplitudes involving Goldstone bosons substituted for externallongitudinal gauge bosons are much easier to calculate.
It has been proven to all ordersin perturbation theory using power counting arguments at least for external gauge bosons.Another possible application of this general concept is to truncate one-loop calculationsinvolving internal gauge bosons to only those diagrams with just Goldstone bosons (no in-ternal gauge bosons or ghosts) [2,3]. This results in a separately finite and gauge invariantsum and is clearly a simpler task than performing the full calculation.
In this short notewe present examples where this equivalence theorem (ET) performs poorly. We find thatthe large logarithms that can appear at one-loop destroy the approximation.
We chooseprocesses that are absent at tree level and first occur at one-loop. In this way we are ableto avoid any subtleties related to renormalization and concentrate on the aspects that arisebeyond tree level but are not specifically related to any renormalization scheme.
We do notargue that this ET is invalid; rather the asymptotic approach to the limit can be gradualand the predicted rates in physically interesting processes can be unreliable.We discuss two processes in the Standard Model that are phenomenologically interest-ing. One is well-known [4] and the other has been considered relatively recently [3].
Firstconsider Higgs decay to two photons, H →γγ. The full one-loop amplitude has been knownfor some time [4], and this process may serve as an interesting theoretical laboratory for theET.
The ET can be employed to calculate the Feynman diagrams involving the gauge bosonsector in the loop. Generic diagrams are shown in Figure 1.
In the full calculation thereare 26 diagrams, while only three diagrams contribute to the ET approximation. We havecalculated these diagrams in the Feynman gauge using the symbolic manipulation programsFORM and MATHEMATICA.
We have used the algorithms for reducing one-loop integralsdeveloped by van Oldenborgh and Vermaseren [5]. This technique gives entirely analytic ex-pressions for the matrix elements.
Gauge invariance is checked analytically for the resultingexpressions.We find that the one-loop decay rate for the W boson loops (not counting fermion loops)2
determined by using the ET isΓET = α2GFM3H16√2π31 + 2M2WC(p1, p2)2 ,(1)The full calculation including all 26 diagrams yields [4]ΓF ULL = α2GFM3H16√2π3ξ11 + 2M2WC(p1, p2)−8M2WC(p1, p2)2 ,(2)whereξ1 ="1 + 6M2WM2H#,(3)and C(p1, p2) is the usual scalar three-point integral with two massless external lines (seebelow) and can be expressed in logarithms alone,C(p1, p2) =12M2Hln2 −z1 −z,(4)withz = 121 +vuut1 −4M2W −iǫM2H. (5)In the very small M2W/M2H limit, C(p1, p2) behaves as 1/(2M2H) ln2(M2W/M2H), and the sub-leading term in ΓF ULL cannot be neglected even for a heavy Higgs boson.
It is perfectlynatural to expect logarithms and dilogarithms to arise in one-loop graphs where integrationover the loop momentum is performed.In Figure 2 we plot the ratio ΓET/ΓF ULL against the Higgs mass MH. Even for a Higgsboson as heavy as 1 TeV, the decay rate has not begun to display the asymptotic behaviorprescibed by the ET.
Eventually the ratio approaches one but only for unrealistically largeHiggs masses.The argument presented so far might be considered only academic since the full calcula-tion for H →γγ is well-known and easily obtained, so we have also explored the effectivenessof the ET in the more complicated process γγ →HH. This process has been suggested as apossible method of measuring the triple-Higgs vertex in the Standard Model.
The ET calcu-lation of the W boson loop contribution has been recently computed [3]. We have performed3
the full one-loop calculation and find the ET calculation to be inaccurate in the region ofinterest. There are 188 one-loop Feynman diagrams in the full calculation compared to 22in the ET approximation.
See Figure 3.We have confidence in our result for the following reasons: (1) We have checked analyt-ically that pµ1Tµν = 0 and pν2Tµν = 0 where Tµν is the polarization tensor for γγ →HH. (2)We have used our program to reproduce published helicity amplitudes for gg →HH (quarkloop) [6], gg →ZZ (quark loop) [7], and the equivalence theorem part of γγ →HH [3].
(3) The simple diagrams in Figure 3 (but not the boxes) were checked versus the programFeynCalc/FeynArts [8].The helicity amplitudes can be expressed in a compact form using the results obtainedin the ET approximation in Ref. [3].
We findMF ULL++= ξ2MET,3a+++ ξ1MET,3b+++ 2−Y D(p1, p3, p2)+2t1C(p1, p3) + 2u1C(p2, p3) + 6sM2Hs1C(p1, p2),(6)MF ULL+−= ξ3MET,3a+−+ 2−Y D(p1, p3, p2) + t21D(p2, p1, p3)+u21D(p1, p2, p3) + 2t1C(p1, p3) + 2u1C(p2, p3) + 2sC(p1, p2),(7)whereξ2 ="1 −4M2WM4HM2H −3M2W + s#,(8)ξ3 ="1 −4M2WM4HM2H −3M2W −s#,(9)and s1 = s −M2H, t1 = t −M2H, u1 = u −M2H, Y = tu −M4H. The matrix elements MET,3a++,MET,3b++and MET,3a+−are those obtained in the ET approximation and given in Ref.
[3] asM0(box), M0(triangle) and −M2(box) respectively (The minus sign is simply a matter ofour convention for the polarization vectors. The M2H in the last line of M2(box) should beM4H.).
The indices 3a and 3b refer to the diagrams in Figure 3. The scalar triangle and boxdiagrams are defined as4
C(p1,p2)=1iπ2Rd4q1(q2−M2W )((q+p1)2−M2W )((q+p1+p2)2−M2W ) ,(10)D(p1,p2,p3)=1iπ2Rd4q1(q2−M2W )((q+p1)2−M2W )((q+p1+p2)2−M2W )((q+p1+p2+p3)2−M2W ) . (11)The momenta of the incoming photons are p1 and p2 while the outgoing Higgs bosons havemomenta p3 and p4.In the limit M2W/M2H →0, M2W/s →0, the factors ξ1, ξ2 and ξ3approach one with only power law corrections.
On the other hand the additional terms donot become negligible immediately.We do not list separately the contributions from the diagrams in Figure 3a and Figure 3bbecause in the full calculation they are no longer separately gauge invariant. The graphs forγγ →H shown in Figure 1 are certainly a gauge invariant set, but once the Higgs is allowedoff-shell as in Figure 3b, a contribution from the graphs in Figure 3a must be included toobtain gauge invariance.
This is not the case for either the subset of diagrams in the ETor for fermion loop contributions. Details of this calculation and issues of phenomenologicalinterest will be presented in a longer paper.In Figure 4 we compare the cross sections given by the full calculation and given by theET.
The two converge in the appropriate limit, but this convergence is quite mild. Thedisagreement is most severe for unequal photon helicities, λ1 = −λ2.
As the center of massenergy increases the approximation gets better, but even at s = 2 TeV there is a substantialdiscrepancy.We have found that large logarithms that arise at one-loop limit the effectiveness of thisET. We believe this behavior is a general feature of such calculations, and one must be carefulnot to place too much confidence in such ET calculations beyond the tree level.
At tree levelwith internal gauge boson lines, this type of logarithm is absent, and the ET should convergequite rapidly to the full result in the appropriate limit. We have not specifically addressedthe issue of one-loop diagrams with the external longitudinal gauge bosons replaced withGoldstone bosons.
We believe large logarithms potentially plague these approximations aswell.5
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FiguresFigure 1: Generic diagrams in the W boson loop contribution to H →γγ. The loopsconsist of all possible combinations of W bosons, Goldstone bosons and ghosts.
The numberof nonzero diagrams is shown, and the number of diagrams in the equivalence theoremcalculation is given in parentheses.Figure 2: Comparison of the full calculation to the equivalence theorem approximationfor H →γγ. The approximation only becomes good for unrealistic Higgs masses.Figure 3: Generic diagrams in the W boson loop contribution to γγ →HH.
Theloops consist of all possible combinations of W bosons, Goldstone bosons and ghosts. Thenumber of nonzero diagrams is shown, and the number of diagrams in the equivalencetheorem calculation is given in parentheses.Figure 4: Comparison of the full calculation to the equivalence theorem approximationfor γγ →HH.
The full calculation is given by solid lines and the ET result by dashed lines. (a) The cross sections for constant center of mass energy s = 1 TeV.
(b) The cross sectionsfor constant center of mass energy s = 2 TeV. (c) The cross section for s = 8M2H for thecase where the two photons have unequal helicity.7
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