physical Refined masses in supersymmetry
Loopfest VI April 17, 2007 Stephen P. Martin Northern Illinois University and Fermilab
I will report on refined calculations of the gluino, squark and Higgs masses in the MSSM beyond leading order.
Based in part on:
hep-ph/0701051 hep-ph/0608026 hep-ph/0501132 (TSIL) with Dave Robertson
Masses are key observables for deciphering SUSY breaking Most of what we do not already know about supersymmetric extensions of the Standard Model involves the soft SUSY-breaking terms with positive mass dimension. Predictions of specific models (Minimal Supergravity, Gauge Mediation, Anomaly Mediation, Extra-dimensional Mediation, ...) allow/require precise calculations.
60 50 40
1
-1
The apparent unification of gauge couplings in the MSSM invites us to
-1
30
2
20 10 0
-1
extrapolate the soft masses up to high scales, to see if they obey some
3
2
-1
Organizing Principle.
6 8 10 12 14 Log10(Q/1 GeV) 16 18
4
What is the Organizing Principle behind SUSY breaking? A reasonable working hypothesis is the Minimal Flavor-Respecting Supersymmetric Standard Model. It is neither too painfully general, nor too naively specific:
Minimal Supergravity (sic) General MSSM 105 new parameters MFRSSM no new flavor or CP violation Gauge-Mediated SUSY Breaking Anomaly-Mediated SUSY Breaking Stuff not thought of yet
MFRSSM parameter count: 3 gaugino masses 5 sfermion (mass)
3 2
M1 , M 2 , M 3 m2 , m2 , m2 , m2 , m2 ~ ~ ~ u ~ e ~ Q L d Au0 , Ad0 , Ae0 , b, m2 u , m2 d (but MZ known) H H Q0
3 (scalar) couplings 3 Higgs mass parameters 1 input RG scale
Total: 15 new parameters beyond the Standard Model
Gaugino Mass Unification is a popular and recurring theme.
M1 (Q) = M2 (Q) = M3 (Q) m1/2
resulting in
at
Q 2 1016 GeV,
M1 : M2 : M3 1 : 2 : 6
for Q near the TeV scale.
To test this, or alternatives to it, we have to relate physical masses to running masses in the Lagrangian (with no superpartners decoupled). Goal: reduce purely theoretical sources of uncertainty to a negligible level, if possible.
Key features of the approach:
To test high-scale organizing principles, work in non-decoupled
SUSY theory with DR (mass-independent) renormalization scheme.
Two-loop diagrams relevant for SUSY involve many comparable
but distinct mass scales simultaneously.
Mass orderings and hierarchies are difficult to anticipate in
advance. So don't try.
Methods should be generic and reusable.
To calculate physical masses Evaluate self-energy = sum of 1-particle irreducible Feynman diagrams:
(s) = (1) (s) + (2) (s) + . . .
where s
= the external momentum invariant.
The complex pole mass
spole = M 2 - iM
is the solution for complex s of:
spole
= =
m2 + (spole ) tree m2 + (1) (m2 ) 1 + (1) (m2 ) + (2) (m2 ) + . . . tree tree tree tree
The pole mass is gauge invariant at each order in perturbation theory, can be related to kinematic masses as measured at colliders.
In the MSSM, squarks, charginos, neutralinos, Higgs scalars can mix:
j k (s) = j
(1)k
(s) + j
(2)k
(s) + . . .
Define functions of the DR masses and couplings:
j j
(1)k
= =
smj +i
lim 2
j
(1)k
(s) + j
(1)k
(2)k
smj +i
lim 2
j
(2)k
(1)k j s
Then the complex pole mass is a gauge-invariant observable:
2 Mj - ij Mj = m2 + j j (1)k
+ j
(2)k
+
k=j
j
(1)k
k
(1)j
/(m2 - m2 ) + . . . j k
However, with tree-level masses in the loop integrals, the kinematics can lead to slow convergence of perturbation theory. . .
For example, the imaginary parts of the one-loop contribution: 2 1 3
will turn on when the DR tree-level masses satisfy:
m1 > m2 + m3
However, the physical width should actually be non-zero only for physical masses satisfying:
M1 > M2 + M3 .
This can be particularly troublesome for the gluino, squarks and h0 , where the tree-level masses differs greatly from the physical masses. In some cases, the computed in the pole mass can even be negative.
To address this, re-expand the pole mass corrections by defining new functions at each loop order L:
(L)k j
= j
(L)k
with all tree-level masses replaced by the real parts of pole masses
The pole mass can then be rewritten as:
2 Mj - ij Mj
= m2 + j j -
k
(1)k
+ j
(1)k
(2)k
+
k=j
j
(1)k
k
(1)j
2 2 /(Mj - Mk )
Re[k
]
(1)j j + ... 2 Mk
This is formally equivalent to the previous expression, up to terms of 3-loop order. It is similar to the on-shell mass or pole mass computed in the on-shell scheme, but with all couplings and mixing matrices computed at tree-level in DR. In the following, I will refer to the first method as "expansion around tree masses", and the method just given as "expansion around pole masses".
Method:
Reduce all self-energies in general theory to a few basis integrals Numerically evaluate basis integrals quickly and reliably for arbitrary masses.
Tarasov's basis and recurrence relations:
S
T
U
M ("Master integral")
Can always reduce 2-loop self-energies to a linear combination of these, with coefficients rational functions of:
s = p2 = external momentum invariant x, y, z, . . . = internal propagator masses
To evaluate basis integrals: Values at s
= 0 are known analytically, in terms of logs, polylogs.
(basis integral) = (another self-energy integral) s = (linear combination of basis integrals)
So, we have a set of coupled, first-order, linear differential equations.
x
Consider the Master integral M (x, y, z, u, v):
y v u
z
Call these 13 integrals In , (n
and the 12 U, S, T basis integrals obtained from it by removing propagators.
= 1, . . . , 13).
Differential equations method for basis integrals
d In = ds
Knm Im + Cn
m
Here Knm are rational functions of s and x, y, z . . ., and Cn are one-loop integrals. These are obtained by using Tarasov's recursion relations. Solve for basis integrals Runge-Kutta complex integration
In using
in the
Im[s]
s-plane, starting from known values at s = 0.
thresholds
Re[s]
TSIL = Two-Loop Self-energy Integral Library
D.G. Robertson, SPM, hep-ph/0501132 Program written in C, callable from C++, Fortran
Basis integrals computed for any values of all masses and s. All subordinate integrals (S, T, U ) for a given master integral (M ) are obtained together in a single numerical computation. Checks on the numerical accuracy follow from changing choice
of contour.
TSIL knows all special cases that have been done analytically
in terms of polylogarithms
Computation times generically 1 second on modern hardware. About 5 to 10 times faster than s2lse, typically.
For the SUSYQCD corrections to the gluino (g ) pole mass, here are the only ~ necessary Master topologies:
0 g ~ g ~ g ~
g ~ g ~ 0 Q g ~ ~ Qj g ~ Q ~ Qk
0 0 g ~
0
(analytic)
g ~
(numerical)
If the first two family squarks are (are not) degenerate, then 12 (18) numerical evaluations by TSIL are required, after taking into account the symmetry. (Note: number of Feynman diagrams is far greater.)
BF S
BF V
MSSF F S
MSF F SF
VF SSSS
These are the Feynman diagram
VSF F F S VF SSF F YF SSS MV SF F S MV F F SF
topologies:
MSSF F V VF SSSV VSF F F V VV F F F S YF SSV ()
Each diagram can be reduced to integrals calculated as part of
VV F F F V MV F F V F MV V F F V VF V V V V YF V V V ()
the Master integral cases on the previous slide.
VF V V F F VF V V SS YF V V S
+ fermion mass insertions + ghost diagrams + counterterms
The same integrals have been applied to obtain the SUSYQCD neutralino and chargino pole mass calculation. (SPM hep-ph/0509115; see also the independent calculation of Robert Schofbeck, next talk.)
~ For the SUSYQCD corrections to the squark (Qj ) pole mass, the necessary
Master topologies are:
~ Qj
0 ~ Qj
~ Qj ~ Qj 0 g ~ g ~ Q
~ Qj
0 0 ~ Qj
0 ~ Qj Q Q ~
~ g Qj
Q 0 g ~ Q
Q
(analytic)
g ~ g ~ ~ Qk
(numerical)
~ Qj
0 ~ Qj
~ Qj
0 ~ Qj
~ Qj
g ~
Q
Assuming small flavor violation, each of the 12 squark pole masses requires 3 (4) numerical evaluations of TSIL, if there is (is not) L-R squark mixing.
Examples of reduction to basis integrals:
x s z v
y u MF F F F S (x, y, z, u, v)
= (xu + yz - vs)M (x, y, z, u, v) - xU (z, x, y, v) - zU (x, z, u, v) -uU (y, u, z, v) - yU (u, y, x, v) + S(x, u, v) + S(y, z, v) +sB(x, z)B(y, u) v y u x = {(v - y - u)U (x, z, u, v) + [A(v) - A(u)]B(x, y)}/(y - z) + (y z)
etc. The same functions are also applicable to Higgs and slepton self-energies.
z
VF F F F S (x, y, z, u, v)
Numerical results for gluino and squark pole masses For simplicity of presentation, take all squarks degenerate and unmixed, and all quarks (including top!) massless.
1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 1 1.05
Msquark/msquark, running
Mgluino/mgluino, running
The "expansion around pole" method is probably most accurate. (The two-loop gluino pole mass in the case of no squark mixing was computed independently and first by Youichi Yamada.)
e e e e e o r p t d d d n n n up u u oe o o rl au arl a .o . pd pt p p xo x en e e ,a , pe p p p oo o o o orlrlr o,, o x x 1l2l 1.. 2
1.00 2
Msquark/Mgluino
e e e e e o r p t d d d n n n up u u oe o o rl au arl a .o . pd pt p p xo x en e e ,a , pe p p p orrr o o o oo o,, o x x 1l2l 1l.l. 2
1 2
Msquark/Mgluino
3
0.95
3
The widths of the gluino and squark pole masses as extracted from the complex pole squared masses M 2
0.015
- iM :
e e e o r p t d d n n u u o o r al a . p p x x e e , p p o, o or o 2l.l 2 e e e o r p t d d n n u u o o r al a . p p x x e e , p p o, o or o 2l.l 2
0.004 0.003
squark/Msquark
0.9
Msquark/Mgluino
gluino/Mgluino
0.010
0.002
0.005
0.001
0.000 1
0.000 1
Msquark/Mgluino
1.1
The "expansion around tree" method fails to give a width consistent with kinematics near the threshold region, as remarked earlier. The "expansion around pole" method is consistent with kinematics, and gives exact agreement with the NLO gluino and squark decay widths found in Beenakker, Hopker, Zerwas hep-ph/9602378.
Three-loop gluino mass corrections for heavy squarks Exploit the fact that beta functions are easier to compute, known to 3-loop order. Let the running parameters in the full MSSM be s , M3 , and in the effective theory with squarks decoupled, s , M3 .
Full MSSM, no decoupling
Msquarks L = ln(
Msquarks M3
2-loop threshold corrections give (s , M3 ) (s , M3 )
)
3-loop s and M3 beta functions known (same as "Split SUSY") pole 3-loop Mg known in terms of s and M3 ~ Standard Model: 4-loop QCD beta function known
Mg ~
Mtop
To obtain the 3-loop contributions for large L
= ln(MQ /Mg ), need: ~ ~
2-loop threshold corrections for M3 in MSSM
(SPM 2006)
2-loop threshold corrections for s in MSSM
(Bern, DeFreitas, Dixon, Wong 2002; Harlander, Mihaila, Steinhauser 2005)
2-loop pole mass in a theory with only fermions
(Gray, Broadhurst, Grafe, Schilcher 1990)
3-loop mass beta function in a theory with only fermions but in different reps
(Tarasov 1982, unpublished, available from KEK server, only in Russian!) In addition, using:
3-loop pole mass in a theory with only fermions (Melnikov and van Ritbergen
1999) but with different reps I obtain the subset of non-log-enhanced contributions that don't involve heavy particle loops, which I expect will be the most important.
Using the effective field theory matching and RG running technique, one obtains
n s Ln n s Ln-1 n s Ln-2
1-loop functions, 0-loop threshold matching 2-loop functions, 1-loop threshold matching 3-loop functions, 2-loop threshold matching
for all n
= loop order. S 3 = 0.163 + 1.112 + 3.74 + 9.62 + ? m2 g ~ 4 = ln(m2 /m2 ). ~ g ~ Q S 3 2 0.16L3 - 1.16L2 - 2.16L + 5.99 + ? Mg = ~ 4 2 2 = ln(MQ /Mg ). ~ ~
For the "expand around tree" method,
2 3-loop Mg , ~
For the "expand around poles" method,
2 3-loop Mg ~
L
Here, "?" means non-log-enhanced contributions from heavy sparticle loops.
Adding these 3-loop effects to the preceding 2-loop results:
1.35 1.30
Mgluino/mgluino, running
1.25
e e e e e o r p t d d d np n n uo u u odl olt o re ao ar a .n pu p p p xe x e, e e ,a prrr p o.. o, o o oxl2l ox o. o 2p 2l2p 2l2, 3 3
2 3
Msquark/Mgluino
1.20 1.15 1.10 1
4
5
The two three-loop approximations are much closer to each other, as expected. The difference between the two-loop and three-loop "expand around pole" results is also quite small.
RG scale dependence of various approximations, fo...