Importance of fermion loops W + W − in elastic scattering
Antonio Dobado, Carlos Quezada-Calonge, Juan José Sanz-Cillero
NNuclear and Particle Physics Proceedings 00 (2021) 1–5
Nuclear andParticle PhysicsProceedings
Importance of fermion loops in W + W − elastic scattering Antonio Dobado, Carlos Quezada-Calonge ∗ , Juan Jos´e Sanz-Cillero Departamento de F´ısica Te´orica and Instituto IPARCOS,Universidad Complutense de Madrid. Plaza de las Ciencias 1, 28040-Madrid, Spain.
Abstract
We test the assumption that fermion-loop corrections to high energy W + W − scattering are negligible when com-pared to the boson-loop ones. Indeed, we find that, if the couplings of the interactions deviate from their StandardModel values, fermion-loop corrections can in fact become as important or even greater than boson-loop correctionsfor some particular regions of the parameter space, and both types of loops should be taken into account. Somepreliminary results are shown. Keywords: E ff ective Theories, Chiral Lagrangians, Beyond Standard Model, Fermion Loops
1. Introduction
When testing for new physics (NP), loop correctionsmust be taken into account to confront experiments withenough precision. A possible place where NP could befound at the LHC is vector-boson scattering. Since nofurther NP states have been found below the TeV scale,the low-energy interactions between Standard Modelparticles accept an e ff ective field theory (EFT) descrip-tion. In particular, if one assumes the existence of ahigh-energy regime where the electroweak (EW) bo-son scattering becomes strongly interacting, the Elec-troweak Chiral Lagrangian (ECL) is the most appropri-ate EFT approach. In addition, the Equivalence Theo-rem (ET) [1] relates, up to corrections O ( M W / √ s ), am-plitudes with longitudinal EW gauge bosons and am-plitudes with would-be Goldstone-bosons in the exter-nal legs. Hence, the ET can be very useful, largelysimplifying computations. In this context, because of ∗ Speaker, corresponding author. ∗∗ Talk given at 23th International Conference in Quantum Chromo-dynamics (QCD 20, 35th anniversary), 27 - 30 October 2020, Mont-pellier - FR.
Email addresses: [email protected] (Antonio Dobado), [email protected] (Carlos Quezada-Calonge ), [email protected] (Juan Jos´e Sanz-Cillero) their formally dominant O ( s ) scaling with the center ofmass energy √ s , only boson-loop corrections are nor-mally considered; as fermion-loop corrections formallyscale like O ( M s ), they are mostly neglected in previ-ous bibliography. Although this assumption is fair inmost cases, we would like to point out in these proceed-ings this is no longer true for some ranges of values ofthe e ff ective couplings, where fermion loops turn nu-merically relevant.The expressions for the fermion-loop contributionsare proportional to the mass of the fermion inside theloop and couplings of the Lagrangian. Some of thesecouplings still allow a 10 % deviation with respect totheir SM values [2]. For this reason it is important totest how relevant are these fermion contributions whenconsidering the whole range of possible coupling val-ues. In this note we will focus on top quark correctionsbecause they are the most significant for our purposes.Further details will be given in a forthcoming article [4].Our results are obtained by using a Higgs E ff ectiveField Theory (HEFT) or Electroweak Chiral Lagrangian(ECL) equipped with a Higgs field [3]. In particular, wefocus on the imaginary part because they enter next-to-leading order in the chiral counting and are not maskedby the purely real lowest order amplitude. The mag-nitude of interest will be the ratio fermions / bosons for a r X i v : . [ h e p - ph ] F e b Nuclear and Particle Physics Proceedings 00 (2021) 1–5 the J = J =
2. Coupling the top quark in the ECL
Neglecting the bottom quark mass M b (cid:28) M t (in addi-tion to all other SM fermions), the lowest order couplingof the top quark to the HEFT is given by the Yukawa likeLagrangian: L Y = −G ( h ) (cid:20) (cid:113) − ˆ ω v M t t ¯ t + i ω v M t ¯ t γ t + − i √ ω + v M t ¯ tP L b + i √ ω − v M t ¯ bP R t (cid:21) . (1)with P R , L = (1 ± γ ) and where we have introduced theHiggs function: G ( h ) = + c hv + c h v + c h v + c h v + ... (2)Thus, at leading order (LO), our e ff ective Lagrangianis given by: L S = F ( h ) ∂ µ ω i ∂ µ ω j (cid:16) δ i j + ω i ω j v (cid:17) ++ ∂ µ h ∂ µ h − V ( h ) + L Y . (3)with the usual HEFT Higgs function F multiplying thewould-be Goldstone ω a kinetic term, F ( h ) = + a hv + b h v + c h v + ... (4)and the Higgs potential, V ( h ) = M h h + d M h v h + d M h v h + ... (5)In the SM case one has a = b = c =
0, for F ( h ), c = c n ≥ = G ( h ), and d = d = d i ≥ in V ( h ).For the one-loop elastic W + W − scattering discussionin this article the only relevant couplings in the LO La-grangian will be a , b , c and d . Finally notice that forthe study beyond the ET, one must also incorporate theEW gauge boson interactions to the ECL [3].
3. Loop corrections to the elastic W + W − scattering At LO, O ( p ), this amplitude T is purely real and itis given by tree-level diagrams made from leading or-der HEFT Lagrangian L vertices. Next contribution T shows up at O ( p ) and consists of a real tree-level part T , tree , from the e ff ective couplings in L (namely a and a ) and one-loop diagrams coming from L ver-tices giving the T , (cid:96) contribution to the O ( p ) amplitudeshowing and imaginary part.Up to the order studied in this work, O ( p ), the realpart of the amplitude is provided by the mentioned threecontributions, Re T = T + T + Re T , (cid:96) . This makesthe study of the NLO one-loop corrections cumber-some. On the other hand, the imaginary part only getscontributions from one-loop diagrams up to this order,Im T = Im T , (cid:96) , and it is fully determined by the LO ef-fective Lagrangian. Therefore, focusing on this imagi-nary part makes the study of importance of fermion cor-rections much simpler and clearer, and thus it will bethe procedure followed in this note. More specifically,we will be studying the imaginary part of the projectedpartial PWA a J ( s ), where: T ( s , t ) = (cid:88) J π K (2 J + P J (cos θ ) a J ( s ) , with K = K =
2) for distinguishable (indistinguish-able) final particles. In the physical energy region, Im a J ( s ) will be obtained from the one-loop absorptive cutsin the s -channel, which we will use as a measure of therelative importance of the various contributions.For scattering amplitudes with only bosons as exter-nal legs it is possible to clearly separate fermion andboson loop diagrams (no mixed loops appear). Now wewill ponder the relevance of each of these two contribu-tions. For this, we use the following notation to refer tothe corresponding absorptive cuts:Fer J = Im [ a J ] t ¯ t , Bos J = Im [ a J ] WW , ZZ , hh ,γγ . (6)Previous bibliography provides the various inter-mediate absorptive cuts: longitudinal vector bosons W + W − [6], longitudinal vector bosons ZZ [7], hh [9]and γγ [8]. For the case of the weak bosons W ± and Z only longitudinal polarizations are being considered inthese proceedings. This is because we are interested inscenarios with a strongly interacting electroweak sym-metry breaking sector where longitudinal componentsdominate the high energy dynamics. In principle, thereis also a contribution from the Zh abortive cut whichvanishes in the ET limit and is neglected in this pro-ceedings.At this point it is important to notice that the relevant Nuclear and Particle Physics Proceedings 00 (2021) 1–5 couplings entering in each PWA are: J = Fer −→ a , c Bos −→ a , b , d J = Fer −→ no dependenceBos −→ a . (7)The goal of this note is to point out that there are re-gions of the parameter space where fermion loops be-come as important as the bosonic loops and thus theyshould not be neglected. In order to explore this possi-bility we will consider the ratio: R J = Fer J Bos J + Fer J ∈ [0 , , (8)since by unitary the imaginary parts of the PWA are al-ways positive. Values of R J close to 0 will indicate thatwe can safely drop fermion loops while significant devi-ations from 0 will point out the relevance of the fermion(top quark) contribution to the W + W − and hh scattering.As it is commonly assumed, we anticipate that fermionloops are negligible in most of the parameter (coupling)space. Nonetheless, we will see that this is not true forsome particular channels and in the region around somespecific values of the parameters.
4. Results for elastic W + W − scattering In this analysis we have explored the couplings in thephenomenological admissible range [2]:0 . ≤ a , b , c , d ≤ . . (9)Concerning the center-o ff -mass energy we have con-sidered the interval: 0 . ≤ √ s ≤ In the following plots we have scanned this region ofthe coupling space for di ff erent values of the di ff erentparameters each time while keeping the others fixed totheir SM values for reference.At this point it is important to state that, when deal-ing with values of the parameters close to the SM, theET is no longer that useful at the energies consideredhere. This is because the SM is a renormalizable the-ory, where the longitudinal components of the elec-troweak gauge bosons are not strongly interacting anddo not play a dominant role [5]. Since the SM would-be Goldstone-boson scattering vanishes in the ET limit √ s (cid:29) M W , one has to go beyond that approximation inthis case. (a) R dependence on a for b = c = d = (b) R dependence on b for a = c = d = Figure 1:
As we see in Figure 1a and Figure 1b, when we scan a and b we can find corrections of 5 % around 0.5 TeV.As the energy increases fermion corrections rapidly be-come irrelevant, as expected.In the c case (Figure 2a), we find corrections ofabout 5% at 0.5 TeV and as the energy increases weget 25 % corrections around 2.5 TeV. The dependenceon d is negligible as we see in Figure 2 b , with at most3 % at low energies.If we do a parameter scan for all values between 0.9and 1.1 for two benchmark energies, 1.5 TeV and 3 TeV,we find that the maximum correction (with R ∼ a = b = d = c = .
9. Thismeans that c is the most sensitive parameter to fermion Nuclear and Particle Physics Proceedings 00 (2021) 1–5 (a) R dependence on c for a = b = c = (b) R dependence on d for a = b = c = Figure 2: corrections . The further c is from 1, the bigger thecorrection is. In this case the ZZ and hh cuts are themost important ones. For the J = does not depend on any parameter apart of the massof the top quark, while the boson part Bos depends onlyon a through the W + W − cut. As it can be seen in Fig-ure 3 we find a wide range of corrections for low energy( 80-90 % at 0.5 TeV for a between 0.95 and 1) and forhigh energies (10-90 % at 3 TeV in the whole range).Thus, in this case the assumption that fermion correc-tions can be neglected does not hold for our study of theimaginary part of the partial waves. Figure 3: R dependence on a . From the previous analysis we can see that, when thevalues of some parameters are below 1, fermion correc-tions can be in fact relevant. This is the case for exam-ple of some NP scenarios like the Minimal CompositeHiggs Models (MCHM) where the parameters dependon the NP scale f . Choosing a value for a = a ∗ we canfind the scale of new physics f via a ∗ = (cid:112) − ξ with ξ = v / f [10]. The parameter b follows the expression b ∗ = − ξ while c ∗ = d ∗ = a ∗ . (a) Ratio for the R PWA in the MCHM (b)
Ratio for the R PWA in the MCHM
Figure 4:
Nuclear and Particle Physics Proceedings 00 (2021) 1–5 As seen in Figure 4b, R has a maximum of 7 % for a = . R the opposite happens: close to the SM values, cor-rections are important both at low and high energy. Forinstance, for a = .
99 fermion corrections vary between90 % and 60 %. Again, the second PWA, a , is moresensitive to fermion loops.
5. Conclusions
In the context of HEFT we have studied the fermionloop contribution to the W + W − → W + W − amplitudeinduced by the top quark and compared it with the bo-son ones. As expected boson contributions dominate inmost of the parameter space. However there are smallregions where fermions become relevant. For the J = R , fermion corrections could be asimportant, or even greater, than the boson ones. As weshowed, the most important parameter for the J = R is c . In this preliminary analysis we find thelargest correction for a = b = d = c = . R , where we have just the a parameter to play with, we find that, even for valuesclose to the SM, fermion corrections are in fact relevant,going all the way up to 90 % for a ∼ R ratio movesfrom 90 % (close to the SM) to 20 % ( a = . R has a maximum of 7 % correction at low energies (0.5GeV) when a = .
9. Therefore R is the most promisingchannel to test fermion loop corrections.In a future work we intend also to include all theintermediate state polarizations (not only longitudinal)and test their relevance, which were assumed negligiblehere [4]. Finally, since we are dealing with the pos-sibility of a strongly interacting electroweak symmetrybreaking sector, one should have to deal with the prob-lem of unitarity of the whole amplitude [11]. Acknowledgements
We would like to thank our collaborators A. Castillo,R. L. Delgado and F. Llanes-Estrada, who participatedin the earlier parts of the research presented in this note[12]. This research is partly supported by the Ministeriode Ciencia e Inovaci´on under research grants FPA2016-75654-C2-1-P and PID2019-108655GB-I00; by theEU STRONG-2020 project under the program H2020-INFRAIA-2018-1 [grantagreement no. 824093]; andby the STSM Grant from COST Action CA16108. C. Quezada-Calonge has been funded by the MINECO(Spain) predoctoral grant BES-2017-082408.
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