Sudakov Effects in Electroweak Corrections

9 Pages·0.12 MB·English

by  P. Ciafaloni

INFNFE-0998
Sudakov effects in
electroweak corrections
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P. Ciafaloni(a) and D. Comelli(b)
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(a) INFN sezione di Lecce, via Arnesano, 73100 Lecce

2 (b) INFN sezione di Ferrara, via Paradiso 12, 44100 Ferrara
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In perturbation theory the infrared structure of the electroweak interactions
ar produceslargecorrectionsproportionaltodoublelogarithmslog2 ms2 ,similarto
Sudakov logarithms in QED, when the scale s is much larger than the typical
mass m of the particles running in the loops. These energy growing corrections
can be particularly relevant for the planned Next Linear Colliders. We study
theseeffectsinthe StandardModelfortheprocesse+e− ff¯andwecompare
→
them with similar corrections coming from SUSY loops.
1 IR divergences: qualitative discussion
Infrared (IR) divergences arise in perturbative calculations from regions of integration over
the momentum k where k is small compared to the typical scales of the process. This is
a well known fact in QED for instance [1] where the problem of an unphysical divergence
is solved by giving the photon a fictitious mass which acts a a cutoff for the IR divergent
integral. Whenreal(bremsstrahlung)andvirtualcontributionsaresummed,thedependence
on this mass cancels and the final result is finite [1]. The (double) logarithms coming from
these contributionsarelargeand,growingwiththe scale,canspoilperturbationtheoryand
need to be resumed. They are usually called Sudakov double logarithms [2]. In the case
of electroweak corrections, similar logarithms arise when the typical scale of the process
considered is much larger than the mass of the particles running in the loops, typically the
W(Z) mass [3, 4, 5]. The expansion parameter results then α log2 s , which is
4sin2θwπ MW2
already 10 % for for energies √s of the order of 1 TeV. This kind of corrections becomes
therefore particularly relevant for next generation of linear colliders (NLC [6]). In the case
of corrections coming from loops with W(Z)s, there is no equivalent of “bremsstrahlung”
like in QED or QCD: the W(Z), unlike the photon, has a definite nonzero mass and is
experimentally detected like a separate particle. In this way the full dependence on the
W(Z) mass is retained in the corrections. Other singularities arise in perturbation theory,
namely those coming from the ultraviolet (UV) region. These divergences can be treated
with the usual renormalization procedure and can be resummed through RGE equations.
However they produce single logs and we expect them to be asymptotically subdominant
with respect to the double logs of IR origin.
We consider here the process e+e− ff¯ in the limit of massless external fermions.
Our notation is that p (p ) is the mom→entum of the incoming e− (e+) and p (p ) is
1 2 3 4
the momentum of the outgoing f (f¯). Furthermore, we define the Mandelstam variables:
s = (p +p )2 = 2p p ,t = (p p )2 = s(1 cosθ),u = (p p )2 = s(1+cosθ).
1 2 1 2 1 − 3 −2 − 1 − 4 −2
In the following we consider only the dominant double logs corrections of IR and collinear
origin coming from one loop perturbation theory and we neglect systematically single logs
(IR, collinear or UV) and “finite” contributions that do not grow with energy. We discuss
the kind of diagrams where we expect these corrections to be present1, and evaluate them
in the asymptotic regime s M2.
≫ w
2 Sudakov logarithms in the vertices
We will consider first as an example, to have a grasp over the effect of the IR double
logs, the “SM-like case” in which a “W boson” having mass M and coupling with fermions
like the photon is exchanged. We take the Born QED amplitude as the reference tree
level amplitude. Then we denote the tree level photon exchange amplitude with =
0
M
i1e2v¯ (p )γ u (p )u¯ (p )γ v (p )andthetreelevelphotonvertexwith = iev¯ (p )γ u (p );
s e 1 µ e 2 f 3 µ f 4 V0 − e 1 µ e 2
e is the electron charge.
Let us first consider IR divergences coming from vertex corrections. Since we work in
the limit of masslessfermions, there is no coupling to the Higgs sector. Moreover,by power
counting arguments, it is easy to see that the vertex correction where the trilinear gauge
boson coupling appears is not IR divergent. The only potentially IR divergent diagram is
thentheoneoffig. 1,whereagaugebosonisexchangedinthet-channel. Itisconvenientto
choosethe momentumofintegrationk tobe the oneofthe exchangedparticle,the bosonin
thiscase. Then,bysimplepowercountingargumentsitiseasytoseethattheIRdivergence
1onlyvertexandboxcorrectionswillbeanalyzed,sincevacuumpolarizationcorrectionsgiveonlysingle
logs,bothofultravioletandinfraredorigin.
1
can only be produced by regions of integration where k 0. The only potentially IR
≈
divergentintegralisthenthescalarintegral,usuallycalledC intheliterature[7]. Anyother
0
integral with k ,k k in the numerator cannot, again by power counting, be IR divergent.
µ µ ν
To understandthe originof the divergences,let us considerthe diagramof fig.1with allthe
masses set to zero. For k 0 the leading term of the vertex amplitude is given by:
≈
α d4k (p p ) α 1 dx 1−x dy
1 2
(1)
V ≈−4πV0Z iπ2 k2(kp )(kp ) ≈−2πV0Z x Z y
1 2 0 0
Wecanseeherethetwologarithmicdivergencesthatarisefromtheintegrationoverthex,y
Feynmannparameters. As is wellknown[1], one ofthem is of collinearoriginandthe other
oneisaproperIRdivergence. Whenwetakesomeofthe externalsquaredmomentaand/or
masses different from zero, they serve as cutoffs for the divergences. Let us consider now
some simple cases that will be useful in the following, where the cutoff is given by a single
scale M. The behavior for C in the asymptotic region s (p +p )2 M2 is as follows:
0 1 2
≡ ≫
1 d4k
C (m ,m ,M,p2,p2,s) (2)
0 1 2 1 2 ≡ iπ2 Z [(k+p )2 m2][k2 M2][(k p )2 m2]
1 − 1 − − 2 − 2
1 s
Re C (0,0,M,0,0,0,s) log2( ) for s M2 (3)
{ 0 } → 2s M2 ≫
1 s
Re C (M,M,M,0,0,0,s) log2( ) for s M2 (4)
{ 0 } → 2s M2 ≫
Then we can use (3) for the vertex of fig. 1 in the asymptotic region finding:
α s
log2( ) for s M2 (5)
V ≈−4πV0 M2 ≫
where we can see the double logarithm behavior of the vertex correction for s M2.
≫
The dependence on the IR logs simply factorizes for the cross section:
1 s dt α s α s
σ 2[1 2 log2 ]=σ [1 2 log2 ]
∝ s Z s |M0| − 4π M2 0 − 4π M2
0
Now let us consider the “susy-like” case in which a fermion is exchanged and a scalar
couples to the external gauge boson (fig 1). In supersymmetry the internal fermion and
scalar, for instance a neutralino and a selectron, have masses of the same order and can,
for our purposes, be taken to have the same mass M. In fact the distinction between the
two masses is irrelevant as long as they are of the same order, since log( s )log( s ) =
m2 M2
log2( s )+log(M2)log( s ) log2( s ) and we are interested only in double logs (single
M2 m2 M2 ≈ M2
logs are neglected). Expression (5) is in this case substituted by:
α d4k M2 s≫M2 M2 α log2( s ) (6)
V ≈−4πV0Z iπ2 k2(kp )(kp ) −→ − s 4πV0 M2
1 2
where we still have the double log behavior coming from the integration over the region
k 0 (remember that always s M2). In this case however, 2p p = s is substituted by
1 2
M≈2 in the numerator, so that th≫e we have log2 s M2 log2 s . In the end the double
M2 → s M2
logarithmbehavior is strongly suppressed by a factor M2 for the SUSY vertex with respect
s
tothe SMcase. This isdue tothe differentcouplingsthatappearinthe vertexcorrections:
fermion-gauge boson coupling in the “SM like” case and fermion-scalar in the “susy like”
case. In the first casethe coupling is, athigh energy,proportionalto pi where i is the label
µ
ofthe externalfermionthe exchangedbosoncouples to. Thenwe havea factorp p where
i j
·
i and j are the fermions connected by the exchanged boson. In the “susy like” case where
scalarsand fermions are exchanged,no such factor is present and p p gets substituted by
i j
·
M2, generally subdominant at high energies.
2
3 Sudakov logarithms in the boxes
Let us consider the exchange of a vector boson of mass M in the s-channel (see fig. 2). In
the limit M 0 and in the IR region, the amplitude is given by:
→
α d4k (p p ) (p p )
1 3 2 4
+ (7)
M≈ 4πM0Z iπ2 (cid:26)k2(p k)(p k) k2(p k)(p k)(cid:27)
1 3 2 4
As is shown schematically in fig. 2, the two terms in this equation come from two different
region of integration. When k 0, then (k+p +p )2 s and we can think the “upper”
1 2
≈ ≈
bosonlinetobeshrunk,likeshowninthefigure. Themirrorsituationisk+p +p 0,k2
1 2
≈ ≈
s. This makes evident the fact that the IR structure of the box is the same of the vertex.
Expression (7) is identical with (1) but with the difference that 2p p = s gets substituted
1 2
by 2p p = 2p p = t. So for SM boxes we have an exchange of s and t variables with
1 3 2 4
− −
respect to SM vertices. In the end for the box contribution in the IR region we can write:
α t
log2( ) (8)
M≈−2πM0 M2
It must be stressed however that this expression is valid only in the asymptotic region
t M2 where the double log behavior is generated, while we assumed s M2.
≫ ≫
Let us now consider the “susy like” box where a scalar particle is exchanged in the
t-channel (see figure 3). In this case the amplitude is:
α d4k M2 M2
+ (9)
M≈ 4πM0Z iπ2 (cid:26)k2(p k)(p k) k2(p k)(p k)(cid:27)
1 3 2 4
Comparingeqs. (7)and(9)wenotethatthesusyamplitudehasafactor M2 withrespect
t
to the SM one. In the IR region t M2 we have, using (4):
≫
α M2 t
log2 (10)
M≈ 4πM0 t M2
Care must be taken when we compute cross sections since, as noted above, eqs. 8) and 10)
arevalidonlywhent M2. Letusthenconsideraregionofthephasespacefromacertain
fixed value of t of ord≫er s on, let’s say s < t < s. Then, if s M2, we can use the
− −2 ≫
expressions valid for t M2. Neglecting unessential factors, the leading box corrections to
≫
the tree level cross sections are given by:
SM ∆σ α −2s dtlog2 t σ αlog2 s
≈ s −s s M2 ≈ 0 M2
SUSY ∆σ α −2sRdtM2 log2 t σ αM2 log2 s
≈ s −s s t M2 ≈ 0 s M2
R
Again, SUSY boxes are depressed by a power factor with respect to SM ones.
To conclude, we expect double logs of IR and collinear origin to give at high energies
large one loop corrections to observables in the SM. This is true both for box and vertex
corrections. On the other hand, in a susy theory, due to the different spins of the particles
exchanged in the loops, these double logs are expected to be power suppressed. For this
reason, in the following we will consider in detail only SM electroweak corrections.
4 Sudakov logarithms in the Standard model
We study the purely electroweak double logarithmic corrections in the Standard Model
coming from the exchange of the W and Z gauge bosons to the process e+e− f¯f in the
→
massless case.
3
For the moment we consider only the massless external fermions µ for leptons, u, c and
d, sforquarks,andweneglect,forthemoment,thebottomquarkwhosecorrectionscontain
a non trivial flavor dependence on the top mass ( future analyses ).
This kind of contributions, as explained before, come from only vertex corrections in
whichonegaugebosonit isexchangedandfromthe boxes(directandcrossed)with twoZs
or two Ws.
The effective vertices γ(Z)f¯f including tree level and dominant double logs are given
by v¯ (p )γ (Vγ(Z)P +Vγ(Z)P )u (p ) with
e 1 µ fL L fR R e 2
Vγ = igs Q (1 1 g2 g2 log2 s 1 g2Qf′ log2 s ) (11)
fL W f − 16π2c2 fL m2 − 16π2 2 Q m2
W Z f W
1 g2 s
Vγ = igs Q (1 g2 log2 ) (12)
fR W f − 16π2c2 fR m2
W Z
and
VZ = i g g (1 1 g2 g2 log2 s 1 g2gf′L log2 s ) (13)
fL c fL − 16π2c2 fL m2 − 16π2 2 g m2
W W Z fL W
g 1 g2 s
VZ = i g (1 g2 log2 ) (14)
fR c fR − 16π2c2 fR m2
W W Z
Hanedregff(ifs′)tLh=e eTx3fte(fr′n)a−l fQerfm(fi′o)sn2Wa.nd f’ its isospin partner. Moreover, gf(f′)R = −Qf(f′)s2W
Defining
v¯ (p )γ P u (p )u¯ (p )γ P v (p ) P P (15)
e 1 µ L,R e 2 f 3 µ L,R f 4 L,R L,R
≡ ⊗
the corrections from box diagrams come from direct and crossed diagrams as a sum of
projected amplitudes on the left-right chiral basis:
B γ P γµP +B γ P γµP +
LL µ L L LR µ L R
⊗ ⊗
B γ P γµP +B γ P γµP
RL µ R L RR µ R R
⊗ ⊗
where
i g4 g2 g2 s+t t
B = ( eL fL(log2 log2 )+
LL s8π2 c4 m2 − m2
W Z Z
1 s+t t
(θ log2 θ log2 ))
4 2f m2 − 1f m2
W W
i g4 g2 g2 s+t t
B = eL fR(log2 log2 )
LR s8π2 c4 m2 − m2
W Z Z
i g4 g2 g2 s+t t
B = eR fL(log2 log2 )
RL s8π2 c4 m2 − m2
W Z Z
i g4 g2 g2 s+t t
B = eR fR(log2 log2 )
RR s8π2 c4 m2 − m2
W Z Z
with the above expressions obtained in the limit s,t M2 and
≫ Z,W
θ =1 for f =µ, d and zero otherwise;
1f
θ =1 for f =ν, u and zero otherwise;
2f
4
Thepositivedoublelogcontributionscomefromthecrossedbox,whilethenegativeones
from the direct diagrams.
Itisclearthattheinterferencebetweenthetwoamplitudes,fortheexchangeofZ bosons,
leads to a depression of the full contribution due to the fact that
s+t t s 1+cosθ
log2 log2 =2log log +finite (16)
m2 − m2 m2 1 cosθ
Z Z Z −
where finite means contributions not increasing as logs. In such a way we lose the leading
log2s factor and we remain with a single log that we neglect. So in leading approximation,
box diagram contributions come only from W exchange.
To obtain the physical observables we must square the full amplitude:
M =M γ P γµP +M γ P γµP + (17)
LL µ L L LR µ L R
⊗ ⊗
M γ P γµP +M γ P γµP
RL µ R L RR µ R R
⊗ ⊗
where
i
M = (Vγ Vγ +VZVZ)+B
LL −s eL fL eL fL LL
i
M = (Vγ Vγ +VZVZ)+B
RL −s eR fL eR fL RL
i
M = (Vγ Vγ +VZVZ )+B
LR −s eL fR eL fR LR
i
M = (Vγ Vγ +VZVZ )+B
RR −s eR fR eR fR RR
and compute the differential cross section
dσ s
= Nf[(M 2+ M 2)(1+cosθ)2+ (18)
dΩ 256π2 c | LL| | RR|
(M 2+ M 2)(1 cosθ)2] (19)
RL LR
| | | | −
with Nf = 1(3) for final state leptons (quarks) and 1 + 2m2Z < cosθ < 1 2m2Z to
c − s − s
be consistent with the above approximations (t m2). In any case we can extend the
≫ − Z
integration region to the full 1 range without modifying the leading results.
±
5 Sudakov logs in the cross section and in the forward
backward asymmetry for e+e f f¯
−
→
We define σ and σ respectively as the tree level (Born) cross section and as the total
B T
cross section containing only the one loop double logarithms . The explicit expressions for
different fermionic final states are given by:
σ /σ (e+e− µµ¯) = 1+( 1.345 +0.282)α 0.330α (20)
T B Box W Z
→ − −
σ /σ (e+e− uu¯) = 1+( 2.139 +0.864)α 0.385α (21)
T B Box W Z
→ − −
σ /σ (e+e− dd¯) = 1+( 3.423 +1.807)α 0.557α (22)
T B Box W Z
→ − −
where α = g2 log2 s 2.710−3log2 s . With the underline “Box” we give the
W,Z 16π2 m2W,Z ≃ m2W,Z
contributions coming from box diagrams, the rest is from vertex corrections.
5
For the forward-backwardasymmetry A (e+e− ff¯) the analytic expressions are:
FB
→
AT /AB (e+e− µµ¯) = 1+( 0.807 +0.770)α 0.002α (23)
FB FB → − Box W − Z
AT /AB (e+e− uu¯) = 1+( 0.521 +0.454)α 0.023α (24)
FB FB → − Box W − Z
AT /AB (e+e− dd¯) = 1+( 0.620 +0.508)α 0.029α (25)
FB FB → − Box W − Z
We see that already at √s = 1(0.5) TeV the parameter α 6(3)10−2 so that the
Z,W
≃
abovecorrectionscan exceedthe ten (six) percent for the crosssections and a resummation
technique (which is under study) is needed.
In the limit α α we can summarize the above results in:
Z W
≃
σ AT
T (µµ¯) 1 1.39α ; FB(µµ¯) 1 0.04α ; (26)
σ ≃ − Z,W AB ≃ − Z,W
B FB
σ AT
T (uu¯) 1 1.66α ; FB(uu¯) 1 0.09α ; (27)
σ ≃ − Z,W AB ≃ − Z,W
B FB
σ AT
T (dd¯) 1 2.17α ; FB(dd¯) 1 0.11α ; (28)
σ ≃ − Z,W AB ≃ − Z,W
B FB
(29)
We can make severalcomments to these results:
Z bosonexchangeisnegative(photon-like)inthevertexcorrections: itdecreasesboth
•
left and right effective vertices. In the boxes, Z exchange contribution does not gives
a double log behavior due to a cancellation between direct and crossed diagrams.
W boson exchange, due to his chiral structure, affects only the left gamma vertex
• proportionally to Qf′ and the left Z vertex to gf′L, giving always contributions
−Qf −gfL
that are positive with respect to the tree level values. Also box diagrams are peculiar
because they affect only the left-left structure of the amplitude and they always give
a negative contribution.
Inσ /σ boxcorrectionsaredominant(morethanthreetimesthevertexones). Since,
T B
•
as noted above, box corrections are given only by W exchange, the e.w. Sudakov
corrections are a peculiar signature of the left-left structure of the full amplitude.
InAT /AB Z correctionsalmostcancel. W contributionsfromvertexareaccidently
• FB FB
almost equal and opposite to the box’s ones leaving a negligible contribution. As a
result,thedoublelogsrelativeeffectismorethanoneorderofmagnitudesmallerthan
for the full cross sections.
Thetotaleffectfromvirtualdouble logsis negativeboth forthe crosssectionsandfor
•
the asymmetries.
6 Conclusions
We have investigated, in one loop electroweakcorrections, the IR origin of double logs that
we denote as e.w. Sudakov corrections. These Sudakov effects can be important for next
generation of colliders running at TeV energies since they grow with energy like the square
ofalogarithm. Insupersymmetricmodels,loopscontainingthe supersymmetricpartnersof
the usual particles do not have double log asymptotical behavior (i.e., the double logs are
6
present but power suppressed). In the SM the e.w. Sudakov corrections are present with a
peculiar chiral structure due to W boson exchange dominance; it should be possible to test
the different chiral contributions with colliders with polarized beams. In any case, already
for TeV machines, proper resummation of such large contributions seems to be needed; in
fact for the various cross sections we find that contributions of order 5-8 % are present
for the planed 500 GeV e+e− NLC [6]. The corrections to the asymmetries considered
in this paper, due to the accidental cancellation between box and vertices contributions,
are almost negligible (one order of magnitude smaller with respect to the cross sections
relative corrections). Sudakov effects in other kind of asymmetries (for instance polarized
asymmetries) and in general in other observables, are currently under study.
Acknowledgments
The authors are indebted to M. Ciafaloni and C. Verzegnassi for clarifying discussions.
A special acknowledgement goes to F. Renard for discussions and a check of some of the
computations.
References
[1] Landau-Lifshits: Relativistic Quantum Field theory IV tome, ed. MIR
[2] V. V. Sudakov, Sov. Phys. JETP 3, 65 (1956)
[3] M. Kuroda, G. Moultaka , D. Schildknecht, Nucl.Phys. B350, 25 (1991)
[4] G.Degrassi, A. Sirlin, Phys. Rev. D46, 3104 (1992)
[5] M.Beccaria,G.Montagna,F.Piccinini,F.M.Renard,C.Verzegnassi;hep-ph/9805250
[6] Physics with e+e− Linear Colliders, ECFA/DESY LC Physics Working Group, hep-
ph/9705442
[7] G. Passarinoand M. Veltman, Nucl.Phys. B160, 151 (1979)
7
p p
2 2
k k
p p
1 1
Figure 1: Vertex diagram in SM (left) and SUSY (right) generating a log2 s . p and p
M2 1 2
are ingoing.
p k p
1 3
IR
= +
⇒
p p
2 4
Figure 2: Box contribution for the SM and effective Feynman diagrams in the IR region
Figure 3: Box contribution for supersymmetry (the crossed diagram is also shown)
8